# Laplace Transforms to solve non IVPs?

## Main Question or Discussion Point

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?

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

HallsofIvy
Homework Helper
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".

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

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?

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".
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.

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".
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.

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.