$y$ be the solution of $y'+y=|x|,x\in \mathbb{R}$, find $y(1)$ Let $y$  be the solution of $$y'+y=|x|,x\in \mathbb{R}$$ $$y(-1)=0$$  Then what is the value of $y(1)$:  Efforts:   I know how to solve the linear equations of general form $$y'+Py=Q$$ where $P$ and $Q$ are continuous function of $x$.   In my case $P=1$ and $Q=|x|$ and both are continuous.   Now  if I apply the formula I get   $$ye^x=\int e^x |x|dx+C$$  Now if I put the initial value I get $$0=\int e^{-1}dx+C$$  Now what should I do? I mean what should be the range of integration and how I proceed from here.  Thanks.
Find the value of $y(1)$ of the ODE $y'+y=|x|$. Let $y$ be the solution of $$y'+y=|x|$$ for $x\in\mathbb{R}$ and $y(-1)=0$. Then the value of $y(1)$ is    $\frac{2}{e}-\frac{2}{e^2}$ $\frac{2}{e}-2e^2$ $2-\frac{2}{e}$ $2-2e$   I don't know what to do with the absolute value function in this problem. So I started like the regular first order equation by calculating integrating factor, and got $$ye^x=\int |x|e^x dx+C.$$ Now I got stuck as how to tackle the absolute value function? Help me to solve this. Thanks!
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