Okay, here is what it looks like:

The problem is stated as:

Set up a system of differential equations with initial conditions to describe the currents $\displaystyle i_2$ and $\displaystyle i_3$ in the electrical network. Assume $\displaystyle E$ is 100V, $\displaystyle R_1$ is 20 ohms, $\displaystyle R_2$ is 40 ohms, L_1 is 0.01 H and $\displaystyle L_2$ is 0.02 H. Currents are initially zero.

My work:

$\displaystyle

E(t) = L_1\frac{di_1}{dt}+i_1R_1$

$\displaystyle E(t) = L_2\frac{di_2}{dt}+i_2R_2$

Kirchhoff's Law

$\displaystyle E(t) = L_1\frac{d(i_2+i_3)}{dt}+(i_2+i_3)R_1$

$\displaystyle E(t) = L_2\frac{di_2}{dt}+i_2R_2$

After this, I can take it from there...but am I setting it up right? How do you set it up?