# Solving differential equations with the laplace transform

### Solving differential equations with the laplace transform

#### Lessons

We can actually use Laplace Transforms to solve differential equations

$L${$y$'} = s $L${$y$} - $y$(0)

And so,

$L${$y$''} = $s^{2}$ $L${$y$} - s $y$(0) - $y$'(0)

And in full generality:

$L${$y^{(n)}$} = $s^{n}$ $L$ {$y$} - $s^{n - 1}$ $y$(0) - $s^{n - 2}$ $y^{'(0)}$ - ... - s $y^{n - 2}$(0) - $y^{n - 1}$(0)

Which can be used to make high level order differential equations much easier to solve, just take the following steps:

1. Convert the differential equation into a Laplace Transform

2. Use the formula learned in this section to turn all Laplace equations into the form $L${$y$}. (Convert all things like $L${$y$''}, or $L${$y$'})

3. Plug in the initial conditions: $y$(0), $y$'(0) = ?

4. Rearrange your equation to isolate $L${$y$} equated to something.

5. Calculate the inverse Laplace transform, which will be your final solution to the original differential equation.

• Introduction
a)
What is $L${$y$'}? What is $L${$y$''}? And how do we use these facts to calculate differential equations using Laplace Transforms?

b)
A brief run-down on the steps used to solve a differential equation by using the Laplace Transform.

• 1.
Calculating Differential Equations Using Laplace Transforms

Solve the initial value differential equation:

$y'' - 3y' + 2y = 6$

With initial values $y$(0) = 2 , $y$'(0) = 6

• 2.
Solve the initial value differential equation:

$y'' - 4y' + 7y = 0$

With initial values $y$(0) = 3, $y$'(0) = 7