The problem goes as follows:
$$\begin{aligned} \frac{d y_1}{dt} &= -ay_1 \\ \frac{d y_2}{dt} &= -by_2 -\frac{dy_1}{dt} \\ y_1(0)&=M \\y_2(0)&=0 \end{aligned}$$
where $y_1,y_2,a,b,M $ are all real and positive. Find the $t$ at which $dy_2/dt$ is at maximum.
This problem is easy enough to solve by just solving for the functions $y_1$ and $y_2$ then finding the solution to $d^2 y/dt^2 =0$.
The answer is $$t = \frac{2\ln(a/b)}{a-b}$$
However, I intuitively think this problem could be more elegantly solved without having to find the functions at all, or rather $y_2(t)$ specifically since finding $y_1(t)$ is trivial: $M\exp(-at)$.
So far I have tried to differentiate the second equation to get
$$\begin{aligned} \frac{d^2 y_2}{dt^2} &= 0\\ &= -b\frac{dy_2}{dt}+a\frac{dy_1}{dt}\\ &= -b(-by_2+ay_1)+a(-ay_1)\\ &= b^2 y_2-y_1(ba+a^2) \\ \therefore \ \frac{y_2}{y_1} &= \frac{a}{b} + \frac{a^2}{b^2} \end{aligned}$$
However, I don't know how much that helps.