As the inductors are in parallel, induced emf across the two inductors is the same, i.e.

$e_1=e_2$

$L_1\left(\frac{d{i}_1}{dt}\right)=L_2\left(\frac{d{i}_2}{dt}\right)$

On integrating both sides, we get
$L_1\int \frac{d{i}_1}{dt}=L_2\int \frac{d{i}_2}{dt}$

$L_1{i}_1=L_2{i}_2$

$\Rightarrow \frac{i_1}{i_2}=\frac{L_2}{L_1}$