# Laplace Differential equation

I have a differential equation that has to be solved with Laplace. I wish someone can provide a full answer

y'' + 4y = x , 0<=x<π
y'' + 4y = πe^-x , π<=x

Initial Conditions:
y(0)=0 y'(0)=1

Integral
Staff Emeritus
Gold Member
Sorry we do not do that here.

Start by applying Laplace tranforms to each of the equations.

Learn this: http://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx
and you will know what to do :)

also your ODE can be written using the step (Heaviside) function:

y'' + 4y = x + H[x-π](πe^(-x) - x)

H[x-π] = 0 at x<π; H[x-π] = 1 at x>=π;

Good Luck.

HallsofIvy
Homework Helper
I have a differential equation that has to be solved with Laplace. I wish someone can provide a full answer

y'' + 4y = x , 0<=x<π
y'' + 4y = πe^-x , π<=x

Initial Conditions:
y(0)=0 y'(0)=1
1) Solve y"+ 4y= x, 0<= x< $\pi$
with initial conditions y(0)= 0, y'(0)= 1.

Evaluate the function, $y_1(x)$, satifying those conditions and its derivative at $x= \pi$ and solve
2) $y''$$+ 4y=$$\pi e^{-x}$
with initial conditions $y(\pi)= y_1(\pi)$, $y'(\pi)= y_1'(\pi)$.

Where I am stuck is how to transform the right part as to write it for the proper laplace transform
How I would do it(and correct me where I am wrong)
y'' + 4y = x[u(x-0)-u(x-π)] + πe^(-x)*u(x-π)

How do you apply the Heavyside? Can you explain me your technique?

Why would you want to use a Heaviside step function? Please advise.

You need to transform the equation. What have you got so far?

HallsofIvy