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

LL{yy'} = s LL{yy} - yy(0)

And so,

LL{yy''} = s2s^{2} LL{yy} - s yy(0) - yy'(0)

And in full generality:

LL{y(n)y^{(n)}} = sns^{n} LL {yy} - sn1s^{n - 1} yy(0) - sn2s^{n - 2} y(0)y^{'(0)} - ... - s yn2y^{n - 2}(0) - yn1y^{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 LL{yy}. (Convert all things like LL{yy''}, or LL{yy'})

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

4. Rearrange your equation to isolate LL{yy} equated to something.

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

  • 1.
    a)
    What is LL{yy'}? What is LL{yy''}? 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.


  • 2.
    Calculating Differential Equations Using Laplace Transforms

    Solve the initial value differential equation:

    y3y+2y=6y'' - 3y' + 2y = 6

    With initial values yy(0) = 2 , yy'(0) = 6


  • 3.
    Solve the initial value differential equation:

    y4y+7y=0y'' - 4y' + 7y = 0

    With initial values yy(0) = 3, yy'(0) = 7