Differential equations

A differential equation is an equation involving a function and its derivative (or derivatives). Our goal is to find the function, if one exists, that satisfies the given differential equation. For example, y = sin(x) is a solution to the ordinary differential equation,



To show this, we can find the derivative of the solution, y'(x), and substitute it, as well as y(x), into the differential equation. Differentiating y = sin(x) yields y' = cos(x). Substituting these into the differential equation,


 
 
 

which satisfies the differential equation. Further, it can be shown that y = Csin(x) is a family of solutions called the general solution of the differential equation, where C is some arbitrary constant. The solution y = sin(x) is referred to as a particular solution to the differential equation.

There are various types of differential equations and a number of different methods that can be used to solve them. It is therefore important to be able to classify different differential equations.

Classifying differential equations

On this page, we will define two types of differential equations: ordinary differential equations and partial differential equations.

An ordinary differential equation (ODE) contains a function with one independent variable along with its derivative(s). A partial differential equation (PDE) contains a function of several variables and their partial derivatives. Below are a few examples of each.

Differential equations can be further classified based on characteristics such as their order, degree, linearity, and whether or not they are homogeneous.

Order and degree

The order of a differential equation is the order of the highest derivative in the equation. Below are some examples of differential equations based on their order.

Given that the differential equation is written in the form of a polynomial and its derivatives, the degree of the differential equation is the power of the highest order derivative in the equation. Below are some examples.

In the first two equations, the degree is determined by y' and y'', both of which have a power of 1. In the third equation, the highest order derivative is , which has a power of 3.

Solving differential equations

There are different approaches to solving differential equations depending on the type of equation being solved. Below are some examples.

Separable differential equations

Given that a differential equation can be written in the form,



where P and Q are functions of x and y respectively, then the functions and their differentials (dx, dy, etc.) can be separated to:



The differential equation can then be solved by integrating both sides with respect to their corresponding variables.

Example

Find the general solution for .

Separating the variables yields:



Then, integrate both sides with their respective variables:


 
 
 
 

Homogeneous differential equations

A first order differential equation in the form is homogeneous if f can be written as some other function, F, such that . The differential equation, can then be solved using the separation of variables technique described in the previous section.

For example, let for the differential equation . Dividing the numerator and denominator of f by x2 yields the equivalent form:


 
 
 
 
 
 

Thus, the differential equation is homogeneous since we can write it as a function of . We can then solve the converted differential equation using separation of variables by letting which results in y = vx and by the product rule. Substituting these new values into the converted differential equation yields:


 
 
 

Separating the variables yields:



Then, integrating both sides of the equation:


 
 

Using back substitution and solving for y, the solution is:


 
 
 
 

Linear differential equations

A differential equation in the form,


,


where P and Q are continuous functions, is a first order linear differential equation. To solve this type of differential equation, multiply both sides by a function u(x), called an integrating factor, such that u'(x) = P(x)u(x). Thus, dividing both sides by u(x) yields:



Then, integrating both sides with respect to x yields the value of u(x):


 
 
 

Using C = 1 makes the process more convenient when multiplying by the integrating factor, so .

Example

Solve

Dividing both sides by sin2(x) yields:


 

The above equation is in the form of a first order linear equation. Since P(x) = csc2(x), the integrating factor is:



Multiplying both sides of the differential equation by the integrating factor yields:


 

By the product rule for derivatives, the left side of the equation above is the derivative of . Substituting this into the equation yields:



Then, integrating both sides and solving for y yields the solution:


 
 
 

Exact differential equation

A differential equation in the form,



where Py(x, y) = Qx(x, y) is called an exact differential equation if there is some function f(x, y) such that:



When these conditions are met, the solution is f(x, y) = C.

Example

Solve

Let P(x, y) = (x2 + 4y) and Q(x, y) = (y4 + 4x). Then,


,


making the differential equation an exact differential equation. To find the solution, integrate either P or Q with respect to their respective variables. Integrating P in terms of x yields:



To find g(y), set fy(x, y) = Q(x, y). Now,


,


so:


 

Then, to find g(y), integrate the derivative with respect to y:


 

Thus:



Since f(x, y) = C,



Combining the constants, the solution is therefore:


 

Alternatively, integrating Q with respect to y yields:



To find h(x), set fx(x, y) = P(x, y). Now,



so:


 

Then, integrating the derivative with respect to x yields:


 

Thus:



Again, since f(x, y) = C,



Combining the constants, the solution is:


 
,

Which agrees with the solution when integrating P with respect to x.