# Explicit check for Laplace transform?

Tags:
1. Jan 14, 2016

### j3dwards

1. The problem statement, all variables and given/known data
Solve the following initial value problem using Laplace transforms: y' + 4y = 3t3 e−4t ; y(0) = 0 . Useful information: Recall that the Laplace transform of y 0 is pY − y(0), where Y is the Laplace Transform of y. The Laplace transform of tk e−at is k!/(p + a)k+1 . Confirm the validity of your result by an explicit check.

2. Relevant equations
tk e−at is k!/(p + a)k+1

3. The attempt at a solution
So I have the solution:

y=3/4 t4e−4t

And I know this is correct.

However is there a specific check I can do to make sure this is correct? ie. What is the explicit check?

2. Jan 14, 2016

### Samy_A

If you understand what a differential equation is, the specific check you could do becomes self-evident.

Said differently: you claim to have the solution. What does it mean that y=3/4 t4e−4t is the solution?

3. Jan 14, 2016

### Ray Vickson

Suppose somebody gave you the alleged solution $y = (3/4) t^4 e^{-4t}$ but did not tell you where it came from; how could you check if it is correct?