Let us assume that we have the following expression:
$\frac{dx'(t,θ)}{dt} = Ax'(t,θ) + B(t)$
where $x'(t,θ) = \frac{\partial x(t,θ)}{\partial θ}$
in which $A$ is constant and $B(t)$ is a function of $t$. Furthermore, $θ$ is a parameter.
How can I solve this differential equation to obtain an expression for $x'(t)$ by using integration techniques? Is there a technique to make this differential equation separable?
I searched the bibliography but haven't managed to found something that can help me.
Every answer is greatly appreciated.
Thank you for your time.
Note that $x=x(t,\theta)$ depends on both $t$ and $\theta$, and we want to find $x'(t,\theta)=\frac{\partial x}{\partial \theta}.$ Let $y=x'(x,t)$, so the equation becomes
$$\frac{dy}{dt}=Ay+B(t)$$ $$\frac{dy}{dt}-Ay=B(t)$$
and multiplying by the integrating factor $e^{-At}$ gives
$$e^{-At}\frac{dy}{dt}-Ae^{-At}y=e^{-At}B(t)$$ $$\frac{d}{dt}\left(ye^{-At}\right)=e^{-At}B(t)$$ $$ye^{-At}=\int e^{-At}B(t)dt+C$$ $$y=e^{At}\int e^{-At}B(t)dt+Ce^{At}$$ $$\implies x'=e^{At}\int e^{-At}B(t)dt+Ce^{At}$$
for an arbitrary constant $C$