Two inductors L₁ and L₂ are connected in parallel and a time varying current flows as shown.

The ratio of currents i₁/i₂ at any time / is

The inductors are in parallel. Therefore, potential difference across them is same.

Hence , V₁ = V₂

$\text{ or }{\mathrm{L}}_1\left(\frac{{\mathrm{di}}_1}{\mathrm{dt}}\right)={\mathrm{L}}_2\left(\frac{{\mathrm{di}}_2}{\mathrm{dt}}\right)$

$\text{ or }{\mathrm{L}}_1\left({\mathrm{di}}_1\right)={\mathrm{L}}_2\left({\mathrm{di}}_2\right)$

Integrating, we get

${\mathrm{L}}_1{\mathrm{i}}_1={\mathrm{L}}_2{\mathrm{i}}_2$

$\text{ or }\frac{{\mathrm{i}}_1}{{\mathrm{i}}_2}=\frac{{\mathrm{L}}_2}{{\mathrm{L}}_1}$