Laplace Transform is difficult. What to do?

Question

I am trying to solve a differential equation that has cumulative normal distributions on the non-homogenous part:

$$ay'' + by' + cy = \gamma e^{mx} \operatorname{erf}(\alpha x + \beta)$$

There are many more terms on the RHS, all of which are functions of erf. The Laplace Transform is easy, but the inverse is not. Matlab is not giving me an answer. Is there something I am missing to study? What can help me deal with these type of "hard" laplace/inverse laplace problems?

Answer 1

In your other question I told you to look at the bilateral Laplace transform of $\text{erf}$ which has a nice closed-form :

for $s \in \mathbb{R}$ and by analytic continuation for $s \in \mathbb{C}$, $\ \ \int_{-\infty}^\infty e^{-x^2/2} e^{-sx}dx = e^{s^2/2}\int_{-\infty}^\infty e^{-(x+s)^2/2} dx= e^{s^2/2}\int_{-\infty}^\infty e^{-x^2/2} dx=e^{s^2/2} \sqrt{2\pi}$. Therefore for $\Re(s) > 0$, $\ \ \int_{-\infty}^\infty (\int_{-\infty}^x e^{-y^2/2}dy) e^{-sx}dx = -\int_{-\infty}^\infty e^{-x^2/2} \frac{e^{-sx}}{-s}dx = \frac{1}{s}e^{s^2/2} \sqrt{2\pi}$

For your ODE, applying partial fraction decomposition to $\frac{1}{s(as^2-bs+c)}$ you'll need to find the inverse Laplace transform of $\frac{1}{s-\alpha}e^{s^2/2}$, expressed in term of $\text{erf}$