# Solve $y'+a(x)y=b(x)$ where $b(x)$ is not continuous

Find all the solutionsof the equation:
$$y'+ay=b(x),\ 0<x<\infty,\$$
where $a$ is a constant and $b(x)=1$ for $0\le x\le \alpha$, and $b(x)=0$ for $x\gt \alpha$ and $\alpha$ here is a positive constant.

Well, I really got stuck on this problems where the right hand side of the differential equation $b(x)$ is not continuous. Could anyone give me some hints on this, please?

• Please clarify, why is $b(x)$ not continuous? – vadim123 May 14 '13 at 3:40
• @vadim123 I am sorry... I just found I copied a wrong problem. I am working on it. – Scorpio19891119 May 14 '13 at 3:42
• @vadim123 Now, it is the problem I need help. – Scorpio19891119 May 14 '13 at 3:45

Solve the two different intervals , $0 \le x \le \alpha$ and $x>\alpha$, separately.
• To piggyback off of this answer.. Once you have the two solutions, you have to stitch them together via continuity at $x = \alpha$. Your solution is necessarily continuous because it is assumed to be differentiable on the half line. – Cameron Williams May 14 '13 at 4:12
Use the method of integrating factors on the corresponding intervals of $b(x)$.