Differential equation for trivial RL series circuit
You will need to solve the differential equation. To do this mathematically yourself you would be on the wrong site. For engineering purposes you could use a CAS. You might use wxMaxima to do the following:

Create a signal by summing some sines:

s(t) := sum((1/(2*n-1))*sin(2*%pi*(2*n-1)*f*t),n,1,3);


Build the equation:

eq: s(t) = R*i + L*'diff(i,t,1);


Use the built-in ODE solver (ode2) to calculate the solutions for the equation:

sols: ode2(eq,i,t);


The result (sols) describes all solutions of the equation, while for a specific case one specific solution is needed. Function ic1() sets boundary conditions that define a single solution. (here t=0 and i=0)

sol: ic1(sols,t=0,i=0);


Build a function out of the solution:

i(t) := ev(rhs(sol));


Set up some component values:

filter1: [
    R=10,
    L=50e-3
];


And plot the result for 50Hz:

wxplot2d([s(t),i(t)],[t,0,0.1]), f=50, filter1;