How to solve the differential equation

Question

I want to solve the following equation:

$\dot{x} = ax + bt$

, where $a$ and $b$ are constants. The solution isn't separable, so I tried guessing without any success so far. Any suggestions?

Answer 1

As commented, this type of problems can be solved systematically by integrating factor, which is a generalization of the following self-explained process:

$$x'-ax=bt\Rightarrow e^{-at}x' - ae^{-at}x = e^{-at}bt\Rightarrow (e^{-at}x)'=be^{-at}t$$

hence $x=e^{at}\int be^{-at}tdt$ where the integral is easy to evaluate with integration by parts.