# Particular maximal solution of the inhomogeneous equation $y^{'} = |t|y + t$ on $\mathcal I = \mathbb R$.

Particular maximal solution of the inhomogeneous equation $y^{'} = |t|y + t$ on $\mathcal I = \mathbb R$.

I've already solved this system in the homogeneous case $b(t)=0$, where I've found the solution (maximal) to be $t \mapsto \eta e^{\int_{t_0}^t |s| ds}$.

However, not I'm looking for a particular maximal solution of the inhomogeneous equation.

Maple gave me the following output:

However, this is by no means useful ? There got to be a way of finding a "nice looking" and "easy to interpret" solution ?

• Sorry, I did a typo. I've now corrected everything. – Shuzheng Feb 17 '15 at 8:30

$\left\lbrace \begin{array} f(x) = e^{\frac{x^2}{2}}+Ce^{-\frac{x^2}{2}} & if & x \leq 0 \\ f(x) = -e^{-\frac{x^2}{2}}+Ce^{\frac{x^2}{2}} & if & x > 0 \end{array} \right.$