Laplace Transforms to solve non IVPs?

1. Jun 8, 2008

Alex6200

Is it possible to use a laplace transform to solve a problem like

x' + x = T

where x is a function of T and x(0) = 5 and x(4) = 7

Or can you only solve initial value problems?

2. Jun 8, 2008

Defennder

Isn't that already an initial value problem? You are given x(0). Yes you can use the Laplace transform to solve it.

3. Jun 9, 2008

HallsofIvy

Staff Emeritus
No, that is not an initial value problem. Since it is a second order differential equation, an "initial value problem" would have x and x' given at the same value of t. Since you are given x at two different values of t, that is a "boundary value problem" which is significantly harder than an initial value problem.

I've always felt that "Laplace transform" was over rated. It gives a very "mechanical" way of solving problems that could be done in other ways. A problem like the one given here, to me, cries out for "variation of parameters".

4. Jun 10, 2008

Defennder

Why is this a second order differential equation? It is x' and not x''.

5. Jun 10, 2008

Alex6200

Allright, my mistake, it should be

x'' + x = T, or something along those lines.

My question is: can I use Laplace transforms to solve endpoint value problems?

6. Jun 10, 2008

Alex6200

I really like Laplace transforms. There's just something really cool and dare I say - transcendental - about how discontinuous areas on the t domain become continuous on the s domain. Although I'd suppose that's true of the integration transform too.

Also, it kind of blows my mind how similar T and sin(T) are on the s-domain.

7. Jun 27, 2008

gamesguru

Agreed, the Laplace transform simply complicates matters more by introducing finding the inverse Laplace transform. There are much easier ways to solve ODE's and simpler ways of reducing PDE's.

8. Jun 28, 2008

alyscia

I think you can. Just leave x'(0) to be an unknown, then you should have a function back that is in terms of T and x'(0). Then apply the fact that x(4) = 7 to find the value of x'(0) which gives you the full equation back.

Laplace transforms, I think, solves exactly the same family of equations that the method of undetermined coefficients solves.