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How to Solve Differential Equations

Last Updated: September 24, 2023 Fact Checked

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 35 people, some anonymous, worked to edit and improve it over time. There are 12 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 2,442,788 times. Learn more...

A differential equation is an equation that relates a function with one or more of its derivatives. In most applications, the functions represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between them.

In this article, we show the techniques required to solve certain types of ordinary differential equations whose solutions can be written out in terms of elementary functions – polynomials, exponentials, logarithms, and trigonometric functions and their inverses. Many of these equations are encountered in real life, but most others cannot be solved using these techniques, instead requiring that the answer be written in terms of special functions, power series, or be computed numerically.

This article assumes that you have a good understanding of both differential and integral calculus, as well as some knowledge of partial derivatives. It is also recommended that you have some knowledge on linear algebra for the theory behind differential equations, especially for the part regarding second-order differential equations, although actually solving them only requires knowledge of calculus.

Preliminaries

{\frac  {{\mathrm  {d}}y}{{\mathrm  {d}}x}}=ky

First Order Equations

y=y(x),

  • If our differential equation is not exact, then there are certain instances where we can find an integrating factor that makes it exact. However, these equations are even harder to find applications of in the sciences, and integrating factors, though guaranteed to exist, are not at all guaranteed to easily be found. As such, we will not go into them here.

Second Order Equations

a

  • We obtain two roots. Because this differential equation is a linear equation, the general solution consists of a linear combination of the individual solutions. Because this is a second-order equation, we know that this is the general solution. There are no others to be found. A more rigorous justification is contained in the existence and uniqueness theorems found in the literature.

W

  • There is yet another way to write out this solution in terms of an amplitude and phase, which is typically more useful in physical applications. See the main article for details on this calculation.

{\frac  {{\mathrm  {d}}^{{2}}x}{{\mathrm  {d}}t^{{2}}}}+3{\frac  {{\mathrm  {d}}x}{{\mathrm  {d}}t}}+10x=0,\quad x(0)=1,\ x'(0)=-1

Repeated roots to the homogeneous differential equation with constant coefficients. Recall that a second-order equation should have two linearly independent solutions. If the characteristic equation yields a repeating root, then the solution set fails to span the space because the solutions are linearly dependent. We must then use reduction of order to find the second linearly independent solution.

y(x)=e^{{rx}}v(x)

  • If the integrals can be done, then one would obtain the general solution in terms of elementary functions. If not, then the solution can be left in integral form.

Characteristic equation. The structure of this differential equation is such that each term is multiplied by a power term whose degree is equal to the order of the derivative.

y(x)=x^{{n}},

Repeated roots. To obtain the second linearly independent solution, we must use reduction of order again.

y=v(x)y_{{1}}

Method of undetermined coefficients. The method of undetermined coefficients is a method that works when the source term is some combination of exponential, trigonometric, hyperbolic, or power terms. These terms are the only terms that have a finitely many number of linearly independent derivatives. In this section, we concentrate on finding the particular solution. [8] X Research source

y_{{c}},

Community Q&A

Ayisha A. Gill

Differential equations relate a function with one or more of its derivatives. Because such relations are extremely common, differential equations have many prominent applications in real life, and because we live in four dimensions, these equations are often partial differential equations. This section aims to discuss some of the more important ones.

{\frac  {{\mathrm  {d}}y}{{\mathrm  {d}}x}}=kx

  • Many differential equations simply cannot be solved by the above methods, especially those mentioned in the discussion section. This occurs when the equation contains variable coefficients and is not the Euler-Cauchy equation, or when the equation is nonlinear, save a few very special examples. The methods above, however, suffice to solve many important differential equations commonly encountered in the sciences. Thanks Helpful 1 Not Helpful 0
  • Unlike differentiation, in which the derivative of any given expression can be calculated, the integral of many expressions simply cannot be found in terms of elementary functions. So do not waste your time trying to integrate an expression that cannot be integrated. Check a table of integrals to verify. The solution of a differential equation that cannot be written in terms of elementary functions can sometimes be written in integral form, but whether the integral can be done analytically is not important in this situation. Thanks Helpful 0 Not Helpful 0

differential equations solving methods

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  • ↑ https://mathinsight.org/ordinary_differential_equation_introduction
  • ↑ https://tutorial.math.lamar.edu/Classes/DE/Linear.aspx
  • ↑ http://web.uvic.ca/~kumara/econ501/schap22.pdf
  • ↑ https://www.mathsisfun.com/calculus/differential-equations-second-order.html
  • ↑ https://tutorial.math.lamar.edu/classes/de/ReductionofOrder.aspx
  • ↑ https://tutorial.math.lamar.edu/classes/de/eulerequations.aspx
  • ↑ https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/17%3A_Second-Order_Differential_Equations/17.02%3A_Nonhomogeneous_Linear_Equations
  • ↑ https://tutorial.math.lamar.edu/classes/de/undeterminedcoefficients.aspx
  • ↑ https://openstax.org/books/calculus-volume-3/pages/7-2-nonhomogeneous-linear-equations
  • ↑ https://mathinsight.org/exponential_growth_decay_differential_equation_refresher
  • ↑ https://mathworld.wolfram.com/BesselDifferentialEquation.html
  • ↑ https://tutorial.math.lamar.edu/classes/de/TheWaveEquation.aspx

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Chapter Contents ⊗

  • Differential Equations
  • Predicting AIDS - a DEs example
  • 1. Solving Differential Equations
  • 2. Separation of Variables
  • 3. Integrable Combinations
  • 4. Linear DEs of Order 1
  • 5. Application: RL Circuits
  • 6. Application: RC Circuits
  • 7. Second Order DEs - Homogeneous
  • 8. Second Order DEs - Damping - RLC
  • 9. Second Order DEs - Forced Response
  • 10. Second Order DEs - Solve Using SNB
  • 11. Euler's Method - a numerical solution for Differential Equations
  • 12. Runge-Kutta (RK4) numerical solution for Differential Equations

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A differential equation (or "DE") contains derivatives or differentials .

Our task is to solve the differential equation. This will involve integration at some point, and we'll (mostly) end up with an expression along the lines of " y = ...".

Recall from the Differential section in the Integration chapter, that a differential can be thought of as a derivative where `dy/dx` is actually not written in fraction form.

Differential Math Problem Solver

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Examples of Differentials

On this page....

Definitions of order & degree General & particular solutions Second order DEs

dx (this means "an infinitely small change in x ") `d\theta` (this means "an infinitely small change in `\theta`") `dt` (this means "an infinitely small change in t ")

Examples of Differential Equations

We saw the following example in the Introduction to this chapter. It involves a derivative, `dy/dx`:

`(dy)/(dx)=x^2-3`

As we did before, we will integrate it. This will be a general solution (involving K , a constant of integration).

So we proceed as follows:

`y=int(x^2-3)dx`

and this gives

`y=x^3/3-3x+K`

But where did that dy go from the `(dy)/(dx)`? Why did it seem to disappear?

In this example, we appear to be integrating the x part only (on the right), but in fact we have integrated with respect to y as well (on the left). DEs are like that - you need to integrate with respect to two (sometimes more) different variables, one at a time.

We could have written our question only using differentials :

dy = ( x 2 − 3) dx

(All I did was to multiply both sides of the original dy/dx in the question by dx .)

Now we integrate both sides, the left side with respect to y (that's why we use " dy ") and the right side with respect to x (that's why we use " dx ") :

`int dy = int(x^2 - 3)dx`

Then the answer is the same as before, but this time we have arrived at it considering the dy part more carefully:

On the left hand side, we have integrated `int dy = int 1 dy` to give us y.

Note about the constant: We have integrated both sides, but there's a constant of integration on the right side only. What happened to the one on the left? The answer is quite straightforward. We do actually get a constant on both sides, but we can combine them into one constant ( K ) which we write on the right hand side.

This example also involves differentials:

`\theta^2 d\theta = sin(t + 0.2) dt`
A function of `theta` with `d theta` on the left side, and A function of t with dt on the right side.

To solve this, we would integrate both sides, one at a time, as follows:

`int theta^2 d theta = int sin(t+0.2)dt` `{theta^3}/3 = -cos(t + 0.2) + K`

We have integrated with respect to θ on the left and with respect to t on the right.

Here is the graph of our solution, taking `K=2`:

Typical solution graph for the Example 2 DE: `theta(t)=root(3)(-3cos(t+0.2)+6)`.

Solving a differential equation

From the above examples, we can see that solving a DE means finding an equation with no derivatives that satisfies the given DE. Solving a differential equation always involves one or more integration steps.

It is important to be able to identify the type of DE we are dealing with before we attempt to solve it.

Definitions

First order DE: Contains only first derivatives

Second order DE: Contains second derivatives (and possibly first derivatives also)

Degree: The highest power of the highest derivative which occurs in the DE.

Example 3 - Order and Degree

a) `(d^2y)/(dx^2)+((dy)/(dx))^3-3x+2y=8`

This DE has order 2 (the highest derivative appearing is the second derivative) and degree 1 (the power of the highest derivative is 1.)

b) `((dy)/(dx))^5-2x=3 sin(x)-sin(y)`

This DE has order 1 (the highest derivative appearing is the first derivative) and degree 5 (the power of the highest derivative is 5.)

c) `(y'')^4+2(y')^7-5y=3`

This DE has order 2 (the highest derivative appearing is the second derivative) and degree 4 (the power of the highest derivative is 4.)

General and Particular Solutions

When we first performed integrations, we obtained a general solution (involving a constant, K ).

We obtained a particular solution by substituting known values for x and y . These known conditions are called boundary conditions (or initial conditions ).

It is the same concept when solving differential equations - find general solution first, then substitute given numbers to find particular solutions.

Let's see some examples of first order, first degree DEs.

a. Find the general solution for the differential equation

`dy + 7x dx = 0`

b. Find the particular solution given that `y(0)=3` .

(a) We simply need to subtract 7 x dx from both sides, then insert integral signs and integrate:

`dy=-7x dx` `intdy=-int7x dx` `y=-7/2x^2+K`

NOTE 1: We are now writing our (simple) example as a differential equation. Earlier, we would have written this example as a basic integral, like this:

`(dy)/(dx)+7x=0`

Then `(dy)/(dx)=-7x` and so `y=-int7x dx=-7/2x^2+K`

The answer is the same - the way of writing it, and thinking about it, is subtly different.

NOTE 2: `int dy` means `int1 dy`, which gives us the answer `y`.

We could also have:

`intdt=t` `intd theta=theta` ` int da=a`

and so on. We'll come across such integrals a lot in this section.

(b) We now use the information y (0) = 3 to find K.

The information means that at x = 0, y = 3. We substitute these values into the equation that we found in part (a), to find the particular solution.

`3=7/2(0)^2+K` gives K = 3.

So the particular solution is: `y=-7/2x^2+3`, an "n"-shaped parabola.

Here is the graph of the particular solution we just found:

Solution graph: `y=-7/2 x^2 + 3`.

Find the particular solution of

given that when `x=0, y=2`.

We can write

as a differential equation:

Integrating both sides gives:

y = 5 x + K

Applying the boundary conditions: x = 0, y = 2 , we have K = 2 so:

y = 5 x + 2

given that:

`y(0) = 3,` `y'(1) = 4,` `y''(2) = 6`

Since y''' = 0, when we integrate once we get:

y'' = A ( A is a constant)

Integrating again gives:

y' = Ax + B ( A, B are constants)
`y = (Ax^2)/2 + Bx + C` ( A, B and C are constants)

The boundary conditions are:

y (0) = 3, y' (1) = 4, y'' (2) = 6

We need to substitute these values into our expressions for y'' and y' and our general solution, `y = (Ax^2)/2 + Bx + C` .

y (0) = 3 gives C = 3 .
y'' (2) = 6 gives A = 6 (Actually, y'' = 6 for any value of x in this problem since there is no x term)
y' (1) = 4 gives B = -2 .

So the particular solution for this question is:

y = 3 x 2 − 2 x + 3

Checking the solution by differentiating and substituting initial conditions:

y' = 6 x − 2 y' (1) = 6(1) − 2 = 4 y'' = 6 y''' = 0

Our solution is correct.

After solving the differential equation,

`(dy)/(dx)ln x-y/x=0`

(we will see how to solve this DE in the next section Separation of Variables ), we obtain the result

Did we get the correct general solution?

Now, if `y=c ln x`, then `(dy)/(dx)=c/x`

[See Derivative of the Logarithmic Function if you are rusty on this.)

`"LHS"=(dy)/(dx)ln x-y/x`

`=(c/x) ln x-((c ln x))/x` `=0` `="RHS"`

We conclude that we have the correct solution.

Second Order DEs

We include two more examples here to give you an idea of second order DEs. We will see later in this chapter how to solve such Second Order Linear DEs .

The general solution of the second order DE

y '' + a 2 y = 0
`y = A cos ax + B sin ax`
y '' − 3 y ' + 2 y = 0
y = Ae 2 x + Be x

If we have the following boundary conditions:

y (0) = 4, y' (0) = 5

then the particular solution is given by:

y = e 2 x + 3 e x

Now we do some examples using second order DEs where we are given a final answer and we need to check if it is the correct solution.

Example 10 - Second Order DE

`y = c_1 sin 2x + 3 cos 2x`

is a general solution for the differential equation

`(d^2y)/(dx^2)+4y=0`

We have a second order differential equation and we have been given the general solution. Our job is to show that the solution is correct.

We do this by substituting the answer into the original 2nd order differential equation.

We need to find the second derivative of y :

y = c 1 sin 2 x + 3 cos 2 x

First derivative:

`(dy)/(dx)=2c_1 cos 2x-6 sin 2x`

Second derivative:

`(d^2y)/(dx^2)=-4c_1 sin 2x-12 cos 2x`

Now for the check step:

`"LHS"=(d^2y)/(dx^2)+4y`

`=[-4c_1sin 2x-12 cos 2x]+` `4(c_1sin 2x+3 cos 2x)` `=0` `="RHS"`

Example 11 - Second Order DE

Show that `(d^2y)/(dx^2)=2(dy)/(dx)` has a solution of y = c 1 + c 2 e 2 x

y = c 1 + c 2 e 2 x , then: `(dy)/(dx)=2c_2e^(2x)`
`(d^2y)/(dx^2)=4c_2e^(2x)`

It is obvious that .`(d^2y)/(dx^2)=2(dy)/(dx)`

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First Order Linear Differential Equations

You might like to read about Differential Equations and Separation of Variables first!

A Differential Equation is an equation with a function and one or more of its derivatives :

Here we will look at solving a special class of Differential Equations called First Order Linear Differential Equations

First Order

They are "First Order" when there is only dy dx , not d 2 y dx 2 or d 3 y dx 3 etc

A first order differential equation is linear when it can be made to look like this:

dy dx + P(x)y = Q(x)

Where P(x) and Q(x) are functions of x.

To solve it there is a special method:

  • We invent two new functions of x, call them u and v , and say that y=uv .
  • We then solve to find u , and then find v , and tidy up and we are done!

And we also use the derivative of y=uv (see Derivative Rules (Product Rule) ):

dy dx = u dv dx + v du dx

Here is a step-by-step method for solving them:

  • 2. Factor the parts involving v
  • 3. Put the v term equal to zero (this gives a differential equation in u and x which can be solved in the next step)
  • 4. Solve using separation of variables to find u
  • 5. Substitute u back into the equation we got at step 2
  • 6. Solve that to find v
  • 7. Finally, substitute u and v into y = uv to get our solution!

Let's try an example to see:

Example 1: Solve this:   dy dx − y x = 1

First, is this linear? Yes, as it is in the form

dy dx + P(x)y = Q(x) where P(x) = − 1 x and Q(x) = 1

So let's follow the steps:

Step 1: Substitute y = uv , and   dy dx = u dv dx + v du dx

Step 2: Factor the parts involving v

Step 3: Put the v term equal to zero

Step 4: Solve using separation of variables to find u

Step 5: Substitute u back into the equation at Step 2

Step 6: Solve this to find v

Step 7: Substitute into y = uv to find the solution to the original equation.

And it produces this nice family of curves:

What is the meaning of those curves?

They are the solution to the equation   dy dx − y x = 1

In other words:

Anywhere on any of those curves the slope minus y x equals 1

Let's check a few points on the c=0.6 curve:

Estmating off the graph (to 1 decimal place):

Why not test a few points yourself? You can plot the curve here .

Perhaps another example to help you? Maybe a little harder?

Example 2: Solve this:   dy dx − 3y x = x

dy dx + P(x)y = Q(x) where P(x) = − 3 x and Q(x) = x

And one more example, this time even harder :

Example 3: Solve this:

  dy dx + 2xy= −2x 3

dy dx + P(x)y = Q(x) where P(x) = 2x and Q(x) = −2x 3

Let's see ... we can integrate by parts ... which says:

∫ RS dx = R ∫ S dx − ∫ R' ( ∫ S dx ) dx

(Side Note: we use R and S here, using u and v could be confusing as they already mean something else.)

Choosing R and S is very important, this is the best choice we found:

  • R = −x 2 and
  • S = 2x e x 2

So let's go:

Put in R = −x 2 and S = 2x e x 2

And also R' = −2x and ∫ S dx = e x 2

And we get this nice family of curves:

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Differential Equation

Differential equations.

In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. Learn how to solve differential equations here.

One of the easiest ways to solve the differential equation is by using explicit formulas. In this article, let us discuss the definition, types, methods to solve the differential equation, order and degree of the differential equation, ordinary differential equations with real-word examples and a solved problem.

Differential Equation Definition

A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable)

dy/dx = f(x)

Here “x” is an independent variable and “y” is a dependent variable

For example, dy/dx = 5x

A differential equation contains derivatives which are either partial derivatives or ordinary derivatives. The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity. There are a lot of differential equations formulas to find the solution of the derivatives.

Order of Differential Equation

The order of the differential equation is the order of the highest order derivative present in the equation. Here some examples for different orders of the differential equation are given.

  • dy/dx = 3x + 2 , The order of the equation is 1
  • (d 2 y/dx 2 )+ 2 (dy/dx)+y = 0. The order is 2
  • (dy/dt)+y = kt. The order is 1

First Order Differential Equation

You can see in the first example, it is a first-order differential equation which has degree equal to 1. All the linear equations in the form of derivatives are in the first order.  It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as:

dy/dx = f(x, y) = y’

Second-Order Differential Equation

The equation which includes the  second-order derivative is the second-order differential equation.  It is represented as;

d/dx(dy/dx) = d 2 y/dx 2 = f”(x) = y”

Degree of Differential Equation

The degree of the differential equation is the power of the highest order derivative, where the original equation is represented in the form of a polynomial equation in derivatives such as y’,y”, y”’, and so on.

Suppose (d 2 y/dx 2 )+ 2 (dy/dx)+y = 0 is a differential equation, so the degree of this equation here is 1. See some more examples here:

  • dy/dx + 1 = 0, degree is 1
  • (y”’) 3 + 3y” + 6y’ – 12 = 0, degree is 3
  • (dy/dx) + cos(dy/dx) = 0; it is not a polynomial equation in y′ and the degree of such a differential equation can not be defined.

Note: Order and degree (if defined) of a differential equation are always positive integers.

Types of Differential Equations

Differential equations can be divided into several types namely

  • Ordinary Differential Equations
  • Partial Differential Equations
  • Linear Differential Equations
  • Nonlinear differential equations
  • Homogeneous Differential Equations
  • Nonhomogeneous Differential Equations

Ordinary Differential Equation

An ordinary differential equation involves function and its derivatives. It contains only one independent variable and one or more of its derivatives with respect to the variable.

The order of ordinary differential equations is defined as the order of the highest derivative that occurs in the equation. The general form of n-th order ODE is given as

F(x, y, y’,…., y n ) = 0

Differential Equations Solutions

A function that satisfies the given differential equation is called its solution. The solution that contains as many arbitrary constants as the order of the differential equation is called a general solution. The solution free from arbitrary constants is called a particular solution. There exist two methods to find the solution of the differential equation.

  • Separation of variables
  • Integrating factor

differential equations solving methods

Separation of the variable is done when the differential equation can be written in the form of dy/dx = f(y)g(x) where f is the function of y only and g is the function of x only. Taking an initial condition, rewrite this problem as 1/f(y)dy= g(x)dx and then integrate on both sides.

Also, check:  Solve Separable Differential Equations

Integrating factor technique is used when the differential equation is of the form dy/dx + p(x)y = q(x) where p and q are both the functions of x only.

First-order differential equation is of the form y’+ P(x)y = Q(x). where P and Q are both functions of x and the first derivative of y. The higher-order differential equation is an equation that contains derivatives of an unknown function which can be either a partial or ordinary derivative. It can be represented in any order.

We also provide a differential equation solver to find the solutions for related problems.

Applications

Differential equations have several applications in different fields such as applied mathematics, science, and engineering. Apart from the technical applications, they are also used in solving many real life problems. Let us see some differential equation applications in real-time.

1) Differential equations describe various exponential growths and decays.

2) They are also used to describe the change in return on investment over time.

3) They are used in the field of medical science for modelling cancer growth or the spread of disease in the body.

4) Movement of electricity can also be described with the help of it.

5) They help economists in finding optimum investment strategies.

6) The motion of waves or a pendulum can also be described using these equations.

The various other applications in engineering are: ­ heat conduction analysis, in physics it can be used to understand the motion of waves. The ordinary differential equation can be utilized as an application in the engineering field for finding the relationship between various parts of the bridge.

Now, go through the differential equations examples in real-life applications .

Linear Differential Equations Real World Example

To understand Differential equations, let us consider this simple example. Have you eve r thought about why a hot cup of coffee cools down when kept under normal conditions? According to Newton, cooling of a hot body is proportional to the temperature difference between its temperature T  and the temperature T 0  of its surrounding. This statement in terms of mathematics can be written as:

dT/dt ∝ (T – T 0 )…………(1)

This is the form of a linear differential equation.

Introducing a proportionality constant k , the above equation can be written as:

dT/dt = k(T – T 0 )    …………(2)

Here, T is the temperature of the body  and t is the time,

T 0 is the temperature of the surrounding,

dT/dt  is the rate of cooling of the body

Differential Equation

Eg:­ dy/dx = 3x

Here, the differential equation contains a derivative that involves a variable (dependent variable, y) w.r.t another variable (independent variable, x). The types of differential equations are ­:

1. An ordinary differential equation ­contains one independent variable and its derivatives. It is frequently called ODE. The general definition of the ordinary differential equation is of the form:­ Given an F, a function os x and y and derivative of y, we have

F(x, y, y’ …..y^(n­1)) = y (n) is an explicit ordinary differential equation of order n.

2. Partial differential equation ­that contains one or more independent variables.

Solved problem

The function given is y = e -3x . We differentiate both sides of the equation with respect to x,

Now we again differentiate the above equation with respect to x,

We substitute the values of dy/dx, d 2 y/dx 2 and y in the differential equation given in the question,

On left hand side we get, LHS =  9e -3x + (-3e -3x ) – 6e -3x

= 9e -3x  – 9e -3x  = 0  (which is equal to RHS)

Therefore, the given function is a solution to the given differential equation.

Differential Equations Practice Questions

  • Find the order and degree, if defined, for the differential equation (dy/dx) – sin x = 0.
  • Verify that the function y = a cos x + b sin x, where, a, b ∈ R is a solution of the differential equation (d 2 y/dx 2 ) + y = 0.
  • Verify that the function y = Ax, where, a, b ∈ R is a solution of the differential equation xy’ = y (x ≠ 0)

Frequently Asked Questions on Differential Equations

What is differential equation.

In Mathematics, a differential equation is an equation with one or more derivatives of a function. The derivative of the function is given by dy/dx. In other words, it is defined as the equation that contains derivatives of one or more dependent variables with respect to one or more independent variables.

Mention the various types of differential equations.

The different types of differential equations are: Ordinary Differential Equations Partial Differential Equations Homogeneous Differential Equations Non-homogeneous Differential Equations Linear Differential Equations Nonlinear Differential Equations

What is the order of the differential Equation?

The order of the highest order derivative present in the differential equation is called the order of the equation. If the order of the differential equation is 1, then it is called the first order. If the order of the equation is 2, then it is called a second-order, and so on.

What is the use of a differential equation?

The main purpose of the differential equation is to compute the function over its entire domain. It is used to describe the exponential growth or decay over time. It has the ability to predict the world around us. It is widely used in various fields such as Physics, Chemistry, Biology, Economics and so on.

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differential equations solving methods

Solving Differential Equations: Implicit Methods

differential equations solving methods

  • Download source code - 859 KB

Image 1

Introduction

One of the hardest things about working on mathematical calculations on the computer is IEEE arithmetical operation. Specifically, it is always an approximation to the actual solutions of the problem, regardless of the problem at hand, whether it is a differential equation or a Taylor series expansion. And how you actually control these errors is very difficult; there is no technique that can deal with every situation, even when the analytical solution is known. This often leads to ad hoc solutions that depend on multiple parameters, and the selection of a solution can also depend on the accuracy and performance you need.

In differential equations, the limit of arithmetical precision happens in so-called stiff differential equations. Under normal Runge-Kutta methods (explicit solutions), the stiff differential equation can give a solution eigenvector that goes rouge, so to speak. This simply means that rounding off errors dominates the solutions and produces the wrong results, so another method for finding the solution is desired. The explicit methods found earlier always had some issues with stability, as there was a limit that would decide when the integration schema would converge. This is referred to as the linear stability domain, where the method will give the correct solutions to the problem, and A-stability is a method that is (nearly) always stable regardless of the step size. In this domain, there is only one method that is A-stable, and that is the backwards Euler method.

In the mathematical world, there is a lot of theory that underpins the usage and stability of different methods. I will not present these here, but literature that gives more detailed explanations is given in the references. Many of the proofs are connected to matrix algebra, so it might not be so easy to follow if you haven't studied that before.

Implicit Methods

Imagine trying to solve an equation of the form:

How would you even attack this problem? We call this differential equation an implicit equation since we essentially have the same value on both sides of the equation and have no obvious method for finding a suitable solution. The general answer to how to actually find a solution is to use what is called a fixed-point iteration. The idea is in the name: you simply use a numerical approximate method that comes closer and closer to the fixed point, in layman's terms, the solution you are actually looking for, after each iteration. For finding these solutions, several mathematical methods exist, but the most widely used is the Newton-Raphson method due to its fast convergence. It is in a family of methods that can generally be constructed by a Taylor series, where the derivation comes from Stoer and Bullirsch's book "Introduction to Numerical Analysis":

This Taylor series is truncated at a certain point, for instance, at the first derivative and yields:

This will give you the classical fixed-point iteration called the Newton-Raphson root-finding method:

One can expand it to include more terms and get an even higher-order method:

which will give the following fixed-point iteration:

I have also seen this method called Schödinger's method, but there does not seem to be a general acceptance of it, at least not for now.

The problem with the higher order method is that you will get multiple roots as the solution, and you will also need the higher and higher order derivatives in order to implement higher and higher order methods.

In this article, I will stick with the simplest possible implementation so it is clear how it can be implemented. Given the derivative and a start value, one can implement the Newton-Raphson method by using the Secant method, i.e., a finite difference approximation:

The real strength of the implicit method is that it is stable for nearly all values of dt , and those timesteps that are not valid can be calculated from the equations a priori (in advance), which none of the explicit methods can do.

To use the Newton-Raphson method, a function that is set equal to zero is needed. Here, the forward Euler method is used for the function:

Moving everything over and setting it equal to zero:

A simple implementation of the method like I showed is not generally recommended since the Newton-Raphson method might not converge to a single value for all functions. Given this fact, a general root-finding method is usually implemented instead. One usually starts with the highest-order method, since this will converge to the solution the fastest. And if that method fails, one simply makes use of a lower-order method. Once all other methods fail, the last and most surefire method, and by far the slowest, is the bisection method, which is used as a last resort.

One often-used implementation of a general and reliable root-finding algorithm is what's called Brent's method. Details on this implementation can be found on Cleve Moler's blog . In short, it combines three methods: the Newton-Rapson implementation with the secant method, the inverse quadratic interpolation method (with three points) used to find a better approximation for the zero crossing point, and at last, the bisection method.

The Newton-Raphson method with the analytical derivative has a degree 2 convergence, but that is usually a bit cumbersome to implement as a general method since it is often not known in advance. Using the secant method eliminates this necessity by implementing the derivative as a finite difference instead, but this comes at a cost since the secant method has a super linear convergence of 1.6 order. But it is way better than the bisection method, which has linear convergence of order 1. For the sake of clarity, a degree 2 method will improve the approximation by two decimal points per iteration, and a degree 1 method will improve the approximation by one decimal point per iteration.

Practical Example

So here is a simple example of how to use the implicit method to solve a simple differential equation:

This example is taken from the Wikipedia article , and as can be seen, the implementation given in my code makes identical graphs to the Wikipedia article.

The code that implements the differential equation and its exact solution are given:

Future Work

This article was just a stepping stone to a complete implementation of the Butcher-Tableau, where we are now aiming to combine explicit and implicit methods like the classical Crank-Nicolson implementation, which combines one explicit step with an implicit step, and this is an example of an IMEX scheme (implicit-explicit).

  • 2 nd January, 2024: Initial posting

This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)

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Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

  • A. H. Tedjani 1 , 
  • A. Z. Amin 2 , 
  • Abdel-Haleem Abdel-Aty 3 , 
  • M. A. Abdelkawy 1,4 , 
  • Mona Mahmoud 5 ,  , 
  • 1. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
  • 2. Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
  • 3. Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia
  • 4. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
  • 5. Department of Physics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia
  • Received: 12 December 2024 Revised: 26 January 2024 Accepted: 06 February 2024 Published: 26 February 2024

MSC : 45Dxx, 65Mxx, 44-xx

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The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.

  • Legendre-Gauss-Lobatto ,
  • fractional Fredholm integro-differential equation ,
  • Caputo fractional derivative ,
  • convergence analysis ,
  • spectral method

Citation: A. H. Tedjani, A. Z. Amin, Abdel-Haleem Abdel-Aty, M. A. Abdelkawy, Mona Mahmoud. Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis[J]. AIMS Mathematics, 2024, 9(4): 7973-8000. doi: 10.3934/math.2024388

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  • This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ -->

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  • Figure 1. The curve of AE as a function of $ \varrho $ in Example 1 is plotted for $ \nu_{1} = 6 $ and $ \alpha_{1} = 0.9 $
  • Figure 2. The curve of AE as a function of $ \varrho $ in Example 1 is plotted for $ \nu_{1} = 12 $ and $ \alpha_{1} = 0.9 $
  • Figure 3. This graph illustrates $ \kappa_{\nu_1} $ for Example 1 at fractional orders $ \alpha_{1} = 0.1, 0.3, 0.5, 0.7, 0.9 $, with $ \nu_{1} = 12 $
  • Figure 4. Convergence for Example 2
  • Figure 5. The AE curve versus $ \varrho $ in Example 3 for $ \alpha = 0.5 $, and $ \nu_{1} = 5 $ and $ \nu_{1} = 8 $, respectively
  • Figure 6. The $ \mathcal{Z}_{Approx}(\varrho) $ and $ \mathcal{Z}(\varrho) $ for Example 3 when for $ \nu_{1} = 5 $ and $ \nu_{1} = 8 $, respectively
  • Figure 7. Convergence for Example 4
  • Figure 8. The approximate solutions for various values of $ \alpha_{1} $
  • Figure 9. The AE curve versus $ \varrho $ in Example 5 for $ \nu_{1} = 12 $ and $ \alpha_{1} = 0.25 $
  • Figure 10. The AE curve versus $\varrho$ in Example 5 for $\nu_{1} = 12$ and $\alpha_{1} = 0.5$
  • Figure 11. The AE curve versus $\varrho$ in Example 5 for $\nu_{1} = 12$ and $\alpha_{1} = 0.75$
  • Figure 12. The AE curve versus $ \varrho $ in Example 5 for $ \nu_{1} = 12 $ and $ \alpha_{1} = 1 $
  • Figure 13. The curve of AE as a function of $ \varrho $ in Example 6 is plotted for $ \nu_{1} = \frac{1}{2} $
  • Figure 14. The $ \mathcal{Z}_{Approx}(\varrho) $ and $ \mathcal{Z}(\varrho) $ for Example 6 for $ \nu_{1} = \frac{1}{2} $

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Remove a code repository from this paper, mark the official implementation from paper authors, add a new evaluation result row, remove a task, add a method, remove a method, edit datasets, new approach method for solving nonlinear differential equations of blood flow with nanoparticle in presence of magnetic field.

25 Feb 2024  ·  Seyed Morteza Hamzeh Pahnehkolaei , Amirreza Kachabi , Milad Heydari Sipey , Davood Domiri Ganji · Edit social preview

In this paper, effect of physical parameters in presence of magnetic field on heat transfer and flow of third grade non-Newtonian Nanofluid in a porous medium with annular cross sectional analytically has been investigated. The viscosity of Nanofluid categorized in 3 model include constant model and variable models with temperature that in variable category Reynolds Model and Vogel's Model has been used to determine the effect of viscosity in flow filed. analytically solution for velocity, temperature, and nanoparticle concentration are developed by Akbari-Ganji's Method (AGM) that has high proximity with numerical solution (Runge-Kutta 4th-order). Physical parameters that used for extract result for non dimensional variables of nonlinear equations are pressure gradient, Brownian motion parameter, thermophoresis parameter, magnetic field intensity and Grashof number. The results show that the increase in the pressure gradient and Thermophoresis parameter and decrease in the Brownian motion parameter cause the rise in the velocity profile. Also the increase in the Grashof number and decrease in MHD parameter cause the rise in the velocity profile. Furthermore, either increase in Thermophoresis or decrease in Brownian motion parameters results in enhancement in nanoparticle concentration. The highest value of velocity is observed when the Vogel's Model is used for viscosity.

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  7. 2.4: Solving Differential Equations by Substitutions

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    In this chapter we will look at solving first order differential equations. The most general first order differential equation can be written as, dy dt = f (y,t) (1) (1) d y d t = f ( y, t) As we will see in this chapter there is no general formula for the solution to (1) (1). What we will do instead is look at several special cases and see how ...

  10. Separable differential equations (article)

    Dividing both sides by 𝑔' (𝑦) we get the separable differential equation. 𝑑𝑦∕𝑑𝑥 = 𝑓 ' (𝑥)∕𝑔' (𝑦) To conclude, a separable equation is basically nothing but the result of implicit differentiation, and to solve it we just reverse that process, namely take the antiderivative of both sides. 1 comment.

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    Second Order Differential Equations - In this chapter we will start looking at second order differential equations. We will concentrate mostly on constant coefficient second order differential equations. We will derive the solutions for homogeneous differential equations and we will use the methods of undetermined coefficients and variation of parameters to solve non homogeneous differential ...

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    To solve the differential equation, we use the five-step technique for solving separable equations. 1. Setting the right-hand side equal to zero gives \(T=75\) as a constant solution. ... a method used to solve a separable differential equation. This page titled 8.3: Separable Differential Equations is shared under a CC BY-NC-SA 4.0 license and ...

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    In this section we solve linear first order differential equations, i.e. differential equations in the form y' + p(t) y = g(t). We give an in depth overview of the process used to solve this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution process.

  18. Solution of First Order Linear Differential Equations

    Here we will look at solving a special class of Differential Equations called First Order Linear Differential Equations. First Order. They are "First Order" when there is only dy dx, not d 2 y dx 2 or d 3 y dx 3 etc. Linear. A first order differential equation is linear when it can be made to look like this:. dy dx + P(x)y = Q(x). Where P(x) and Q(x) are functions of x.. To solve it there is a ...

  19. Numerical Methods

    Many differential equations cannot be solved exactly. For these DE's we can use numerical methods to get approximate solutions. In the previous session the computer used numerical methods to draw the integral curves. We will start with Euler's method. This is the simplest numerical method, akin to approximating integrals using rectangles ...

  20. Differential Equations (Definition, Types, Order, Degree, Examples)

    One of the easiest ways to solve the differential equation is by using explicit formulas. In this article, let us discuss the definition, types, methods to solve the differential equation, order and degree of the differential equation, ordinary differential equations with real-word examples and a solved problem.

  21. Methods of Solving Differential Equations

    Methods of Solving Differential Equation: A differential equation is an equation that contains one or more functions with its derivatives. It is primarily used in physics, engineering, biology, etc. The differential equation's primary purpose is to study solutions that satisfy the equations. The solution of the differential equation is the ...

  22. Solving Differential Equations: Implicit Methods

    Practical Example. So here is a simple example of how to use the implicit method to solve a simple differential equation: dy dt = −y2 d y d t = − y 2. This example is taken from the Wikipedia article, and as can be seen, the implementation given in my code makes identical graphs to the Wikipedia article.

  23. Proper orthogonal decomposition method of constructing ...

    Proper orthogonal decomposition method of constructing a reduced-order model for solving partial differential equations with parametrized initial values. Author links open overlay panel Yuto Nakamura, Shintaro Sato, Naofumi Ohnishi. Show more. ... the choice of orthogonal basis for the Galerkin projection affects the method of solving these ...

  24. 8.02: Euler's Method for Solving Ordinary Differential Equations

    Euler's method is a numerical technique to solve first-order ordinary differential equations of the form. dy dx = f(x, y), y(x0) = y0 (8.2.1.1) Only first-order ordinary differential equations of the form given by Equation (8.2.1.1) can be solved by using Euler's method. In another lesson, we discuss how Euler's method is used to solve ...

  25. PDF New approach method for solving nonlinear differential equations of

    method. To solve differential equations, there are simple, accurate methods known as weighted residual methods (WRMs). Collocation, Galerkin and Least Square are examples of the WRMs. In this context Hatami et al. [19] studied the third grade non-Newtonian blood conveying gold nanoparticles in a porous and hollow vessel by two analytical ...

  26. Solving differential‐algebraic equations in power system dynamic

    1 INTRODUCTION. Solving differential-algebraic equations (DAEs) is a fundamental task for time-domain simulation in the power system dynamic analysis where fast computation time and accurate solutions are required [].These DAEs include a set of ordinary differential equations (ODEs) modeling the dynamics of synchronous generators, exciters, and governors, along with nonlinear algebraic ...

  27. Legendre spectral collocation method for solving nonlinear fractional

    <abstract> The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs.

  28. New approach method for solving nonlinear differential equations of

    New approach method for solving nonlinear differential equations of blood flow with nanoparticle in presence of magnetic field ... and nanoparticle concentration are developed by Akbari-Ganji's Method (AGM) that has high proximity with numerical solution (Runge-Kutta 4th-order). Physical parameters that used for extract result for non ...