Solving initial value problems with the Laplace transform

Remarks

Example: for $y'' + y = 0$, $y(0) = 0$, $y'(0) = 1$, the transform gives $s^2 Y(s) - 1 + Y(s) = 0$. So $Y(s) = 1/(s^2 + 1)$ and $y(t) = \sin t$. For $a_n y^{(n)} + \cdots + a_0 y = g(t)$ with zero initial values, $Y(s) = G(s) / P(s)$, where $P(s) = a_n s^n + \cdots + a_0$ and $G$ is the transform of $g$. The convolution theorem gives $y = h * g$ with $h = \mathcal{L}^{-1}\{1/P\}$. The method handles discontinuous forcing, such as a step function, and impulses, such as a Dirac delta.