Outside Temperature- First-Order, Linear, Initial Value Differential Equation

Question

I am trying to solve a first-order initial value differential equation. Here is the equation:

Is this separable? I feel like you could solve it using an integrating factor where P(x)=k and Q(x) is equal to everything on the right side of the equation. But is P(x) allowed to be a constant? I can't seem to get the right answer.

Answer 1

Write the equation $\dot{u}(t)+k u(t) = C(t)$, then $e^{kt}(\dot{u}(t)+k u(t)) = e^{kt}C(t)$ and letting $w(t)=e^{kt} u(t)$ we see that $\dot{w}(t) = e^{kt}C(t)$, which is just an integration.