Category Archives: Laplace Transform

Solve IVPs using Laplace Transform – a problematic example

January 12, 2024 – 8:06 pm

Laplace transform has been used to solve Initial Value Problems. This is a topic covered in a standard differential equation course at most universities. In this post I will describe a strange situation: the solution of 2nd order ODE obtained by using Laplace transform appears to violate one of the two initial conditions. Then a short justification will be given. It turns out to be a very simple problem, just one may get caught off guard.

Background. Consider a 2nd order linear constant coefficient differential equation:
y'' + p y' + q y = f(t), y(0)=a, y'(0)=b.
Applying Laplace transform, we get
(s^2 Y(s) - sa -b) + p(sY(s) -a) +q Y(s) = F(s), where the capital-case functions are the Laplace transforms of the corresponding lowercase functions: F(s)=L[f(t)] and Y(s)=L[y(t)]. It follows that
(s^2 + p s + q)Y(s) = F(s) + a s + pa, or equally, Y(s) =\frac{F(s) + a s + pa}{s^2 + p s + q}.
Applying the inverse transform, we obtain the solution of the IVP.

Laplace Transform of the delta functions. Let \delta_c(t) be the delta function at c\ge 0. We know that L[\delta_c(t)] = e^{-c s}. In particular, L[\delta_0(t)] = 1.

A problematic example. Consider the IVP y''+y=\delta_0(t), y(0)=0, y'(0)=0.
Laplace Transform: (s^2+1)Y(s) = 1, or equally, Y(s) = \frac{1}{s^2+1}.
Inverse Transform: y(t)= L^{-1}[Y(s)] = \sin t, t\ge 0.

The above process is fairly straightforward. Most of us would stop here and move onto the next problem. Some of you may double check:
(1) y'' + y =0 for t \neq 0 , where the delta function \delta_0(t) also vanishes;
(2) y(0) = \sin 0 =0, the first initial condition checked;
(3) y'=\cos t, y'(0)=\cos 0 =1, wait, what? — it violates the second initial condition.

You can check the above computation. There is no error. How is it a solution since it does not satisfy the initial conditions that we started with? Should we call it a solution? Is Laplace transform not applicable here? These were my initial reactions.

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By Pengfei | Tagged , , , | Comments (0)