# Differential equation for RLC circuit

I sit all day on a simple task and I still can't solve it. In the picture you can see the circuit it is about. I have to do the differential equation and solve it in a way that I can determine the voltage at the capacitor Uc(t). I have already solved RLC circuits, but I have problems with the parallel circuit between L2 and R3, which confuses me a lot. Can someone please help me? https://i.stack.imgur.com/TTxKu.jpg

This is my current approach: https://i.stack.imgur.com/zAiA1.jpg

My problem is that I can't just divide by L1 and L2. Using i_L2 = i - i_R3 doesn't give me a result either. So all my steps following the approach were/are unfortunately wrong.

• So I assume you have written down the Kirchhoff's law for the circuit. What was the result you obtained? – Matti P. May 24 at 12:23
• Hi! Thanks for the quick answer. I have inserted my current approach in the question above, unfortunately, I do not yet know how to insert a picture in a comment. – Noah May 24 at 13:03

You can use current divider law and we have $$i_{R_3}=\frac{i\omega L_2}{\sqrt{(R_3)^2+(\omega L_2)^2}}$$ or $$i_{L_2}=\frac{iR_3}{\sqrt{(R_3)^2+(\omega L_2)^2}}$$ where $$\omega$$ is angular frequency of the circuit. Plugging any of the two into your equation will yield a differential equation in $$i,t$$ which I think should be solvable.Note that $$i_{L_2}$$ will be complex part of the current $$i$$ . I have ignored that as calculations can be done with real part too.
$$i_{L2}+i_{R3}=i$$ $$R_3 \, i_{R3}= L_2 \, d/dt\, i_{L2}$$