Consider the differential equation $$ y' = a(t)y + b(t). $$ The general solutions takes the form $y : \mathbb{R} \rightarrow \mathbb{R}$, $$ (1) \qquad y(t) = w(t, 0)c_0 + \int_{0}^t w(t,s) b(s) ds, $$ where $w(t,s) = e^{\int_s^t a(\tau) d\tau}$ is such that $y(t) = w(t,s)c_s$ is a solution to $y' = a(t)y$ given $y(s) = c_s$.
Question. How can we physically interpret the solutions $(1)$ in terms of $w$ and $b$? How can this solution be guessed from simple principles?
Linearity principle: If $b(t) = b_1(t) + b_2(t)$, then the solutions $y_i$ to $y' = a(t)y + b_i(t)$ are such that $y = y_1 + y_2$ is a solution to $y' = a(t)y + b(t)$.
Let $y$ be the amount of money in a bank account (or $n$ bank account) with a continuously variable interest rate that is, at each time $t$, a linear function of $y$. Furthermore, money is added and withdrawn from the account following $b(t)$: in any time interval $I$, $\int_I b(t) dt$ is the amount added during the time interval $I$. Thus $y$ satisfies the differential equation $y' = a(t) y + b(t)$. At time $t=0$, we suppose $y(t) = 0$ (money is only added through $b$).
We can simplify the situation by modelling the money input as discrete chunks. Thus, in the interval $[0,t]$, we write $$ b(t) = \sum_{i=1}^n b_i(t), \quad b_i(t) = b(t) \chi_{R_i}(t), $$ where $R_i = \left[\frac{i-1}{n}, \frac{i}{n}\right]$. The solutions $y_i$ to $y' = a(t) y + b_i$ are approximately the solutions to $y' = a(t)y$ given $y\left(\frac{i}{n}\right) = \frac{1}{n}b\left(\frac{i}{n}\right)$. Thus $$ y_i(t) \approx \frac{1}{n}w\left(t, \frac{i}{n}\right) b\left(\frac{i}{n}\right). $$ By the "linearity principle", the solution to $y' = a(t)y + b(t)$ is then $$ y(t) \approx \frac{1}{n}\sum_{i=1}^n w\left(t, \frac{i}{n}\right)b\left(\frac{i}{n}\right). $$ Letting $n \rightarrow \infty$ we can guess that $$ y(t) = \int_{0}^tw(t,s) b(s) ds. $$