Explicit check for Laplace transform?

[j3dwards]

Homework Statement

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 t^k e^−at is k!/(p + a)^k+1 . Confirm the validity of your result by an explicit check.

Homework Equations

t^k e^−at is k!/(p + a)^k+1

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?

 

[Samy_A]

Homework Statement

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 t^k e^−at is k!/(p + a)^k+1 . Confirm the validity of your result by an explicit check.

Homework Equations

t^k e^−at is k!/(p + a)^k+1

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?

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?

 

[Ray Vickson]

Homework Statement

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 t^k e^−at is k!/(p + a)^k+1 . Confirm the validity of your result by an explicit check.

Homework Equations

t^k e^−at is k!/(p + a)^k+1

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?

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?