Hi,

I'm trying to learn electronics and math in my spare time, and now I'm stuck somewhere

with electronics because my math isn't quite that good yet.

View attachment 28281
You are given that \(\displaystyle L\frac{di(t)a}{dt}+ R_1i(t)+ v(t)= 0\) and that \(\displaystyle i(t)= C\frac{dv(t)}{dt}+ \frac{v(t)}{R_2}\) and are told to replace i(t) in the first equation by its form in the second equation. That will give you \(\displaystyle L\frac{d(C\frac{dv(t)}{dt}+ \frac{v(t)}{R_2})}{dt}+ R_1i(t)+ v(t)= 0\). Yes, you now have to differentiate that expression- and since that expression already has a derivative the result will have a second derivative. That will give \(\displaystyle LC\frac{d^2v(t)}{dt^2}+\frac{1}{R_2}\frac{dv(t)}{dt}+ R_1C\frac{dv(t)}{dt}+ \frac{R_1}{R_2}v(t)+ v(t)= 0\).