So In my research I stumbled upon a difficult ODE, It goes like this $$ y''(x)-[(\operatorname{Heaviside}(ax)+b]y(x)=0, $$ (a,b are the respective constants) I tried approximating Heaviside with analytical terms, which gave me these ODE's
$$ y''(x)-[b/2*(1+2/π * \arctan(x/ϵ) )+a]y(x)=0, $$ or
$$ y''(x)-[a/(1+e^x\epsilon]+b)y(x)=0$$
The epsilon should multiply the power as well, it doesn't work for some reason. The mighty wolfram gives me answers that I barely understand, How would you suggest I go about it? Should I try a different approach? is it even solvable? Laplace transform also didn't help me. Thanks a lot!
Assumptions. $a$ is positive and initial conditions $y(x_{0})$ and $y^{\prime}(x_{0})$ are specified at $x_{0}<0$.
As John Barber points out, $H(ax)=H(x)$ in this case. Therefore, the ODE is equivalent to $$ y^{\prime\prime}(x)=\begin{cases} \left(b+0\right)y(x) & \text{if }x<0\\ \left(b+1\right)y(x) & \text{if }x>0. \end{cases} $$ It follows that $$ y(x)=c_{1}e^{x\sqrt{b}}+c_{2}e^{-x\sqrt{b}}\qquad\text{for }x_{0}<x\leq0 $$ where $c_{1}$ and $c_{2}$ are obtained by solving the linear system $$ \begin{pmatrix}\phantom{\sqrt{b}}e^{x_{0}\sqrt{b}} & \phantom{-\sqrt{b}}e^{-x_{0}\sqrt{b}}\\ \sqrt{b}e^{x_{0}\sqrt{b}} & -\sqrt{b}e^{-x_{0}\sqrt{b}} \end{pmatrix}\begin{pmatrix}c_{1}\\ c_{2} \end{pmatrix}=\begin{pmatrix}y(x_{0})\\ y^{\prime}(x_{0}) \end{pmatrix}. $$ Similarly, $$ y(x)=C_{1}e^{x\sqrt{b+1}}+C_{2}e^{-x\sqrt{b+1}}\qquad\text{for }x>0 $$ where $C_{1}$ and $C_{2}$ are obtained by solving the linear system $$ \begin{pmatrix}1 & 1\\ \sqrt{b+1} & -\sqrt{b+1} \end{pmatrix}\begin{pmatrix}C_{1}\\ C_{2} \end{pmatrix}=\begin{pmatrix}y(0)\\ y^{\prime}(0) \end{pmatrix}. $$