# Solving differential equations with step functions

### Solving differential equations with step functions

#### Lessons

We can use Laplace Transforms to solve second order differential equations with step functions

The following formulas may be needed:

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

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

$L${$u_{c}(t)$$f(t - c)$} = $e^{-sc}$$L${$f(t)$}

• 1.
Solving Differential Equations with Step Functions Using the Laplace Transform

Solve the following differential equation:

$3y'' - 2y' + 2y = 2 - 2u_{2}(t)$

Using initial conditions $y$(0) = 3, $y$'(0) = 2