# Differential equation from a Wheatstone bridge with Inductor

I've been given the following circuit and been asked to give the differential equation showing the relationship between $u_e(t)$ and $u_a(t)$ The differential equation should be in the form $$u_{a}^{\prime}(t) + k_1\cdot u_{a}(t)= k_2\cdot u_{e}(t) + k_3\cdot u_{e}^{\prime}(t) + k_4\cdot u_{e}^{\prime\prime}(t)$$

Some of the coefficients can be zero.

I can see that I'm going to have to integrate the inductor term in order to get an expression for the current, but I'm getting really confused about where I need to start and how to proceed.

I tried applying Kirschoff's laws as far as I could but I don't see how to fit them into each other to get a meaningful result.

Voltage at left voltage divider (between R1 and R2) is just input voltage scaled down by a constant factor $\frac{R_2}{R_1+R_2}$.
Voltage at right terminal (between R3 and R4) is R4 times current through inductor $I_L$. Get current through inductor by setting up an appropriate diffenrential equation using $U_L=-L \frac{dI_L}{dt}$ and KVL and Ohm's law.