Solving Differential Equations Examples

Find a solution to the differential equation

y' = 2x.

The expression 2x looks like it came from using the power rule get 2x after taking the derivative, we could have started with

y = x².

This is, of course, only one possible solution. There are infinitely many other solutions.

Find all solutions to the differential equation

y' = 2x.

Before taking the derivative, we could have started with

y = x².

We could also have started with

y = x² + 1

or

y = x² - 9

or

y = x² + π.

No matter what constant we stick on after x², we'll still have a solution to the differential equation.

This means we can write

y = x² + C,

where C can be any constant, to take care of all solutions at once!

At some point in the future, the constant + C in the example above will be known as the constant of integration.

Find a solution to the differential equation

Any solution to this d.e. must be its own derivative. The most obvious such function is

y = e^x.

But we could also have

y = 5e^x,

since then y' = 5e^x also. Or we could have

y = -4e^x.

Any equation of the form

y = Ce^x

(where C is a constant) is a solution to this differential equation.

In the example above, the constant C can be anything, including zero. If C = 0 then y is a constant function;

y = 0e^x = 0

Since the derivative of the constant function y = 0 is also 0,
the differential equation

is satisfied.

Find a solution to the d.e.

y" = 5.

Here we need to think backwards twice. What was y'? We took the derivative of y' and got 5, so we could have started with

y' = 5x

If the derivative of y is 5x, then we could have started with

Find all solutions to the d.e.

y" = 5.

Thinking backwards the first time, we have

y' = 5x + B

where B is any constant. Thinking backwards again, what function has a derivative of 5x + B? One answer is

However, we could also add on any constant to this expression, since that wouldn't change the function's derivative. So the solution can be any function of the form