## Differential Equations

An equation involving derivatives or a derived function is called a differential equation.

Differential equations are solved using integration.

When a differential equation is solved, there are two types of solution :

general solution which will involve arbitrary constants. This can lead to solutions which are families of curves.

or

particular solution where information is given allowing the values of the arbitrary constants to be determined.

Two types of differential equations are studied: first order or second order:

 Type of differential equation Notation Example First order contain f '(x), y ' or dy⁄dx dy⁄dx = 3x − 2 Second order contain f ''(x), y '' or d2y⁄dx2 y '' + y ' = 2x

In this and the next topic all three types of notation will be used.

### Verifying a Given Solution

Sometimes the solution to a differential equation is provided and has be be verified. It is important to set out this type of problem in an orderly and mathematically correct way.

Example

Given that y '' + y ' = 2x, verify that y = (x − 1)2 is a particular solution.

Answer (LHS means left hand side and RHS means right hand side.)

 To show that y = (x − 1)2 is a solution of y '' + y ' = 2x Let y = (x − 1)2 ⇒ y ' = 2x − 2 ⇒ y '' = 2 Thus, LHS = y '' + y ' = 2 + 2x − 2 = 2x = RHS Hence y = (x − 1)2 is a particular solution.

### General Solutions

Differential equations are solved by integrating both sides of the equation.

Example

Find the function g(x) given that g '(x) = 4 − 2x

Solution

Integrating both sides g(x) = 4x − x2 + c

This general solution produces a family of curves. ### Particular Solutions

A method of solving differential equations is known as separating the variables. The derivative dydx is not a fraction but can be treated in a similar manner when solving differential equations.

Once solved the value of the arbitrary constant can be found if enough information is given.

Example

Solve the differential equation dydx = 2x given that y = 26 when x = 5

dy = 2x. dx       (separating the variables)

∫ dy = ∫ 2x.dx        (integrating both sides)

y = x2 + c

When x = 5, y = 26

26 = 25 + c

c = 1

The solution is y = x2 + 1

### Second Order Differential Equations

Second order differential equations are equations containing terms such as f ''(x), y '' or d2ydx2

Solving this type of differential equation involves integrating twice.

Example

Find the general solution to the differential equation s ''(t) = -9.8

Integrating both sides:

∫ s ''(t) dt = ∫ -9.8 dt

s '(t) = -9.8t + A

Integrating both sides again:

∫ s '(t) dt = ∫ (-9.8t + A)dt

s(t) = -4.9t2 + At + B