1. You could "guess" the form of the answer. e.g. i = A sin (mt) + B cos (mt)
2. Calculate the derivative and second derivative of your guessed i
3. Substitute for i and its derivatives in the differential equation
4. Choose A, B and m to make the equation work
The last bit can be done by choosing special values for t such as t= 0 and/or mt = pi/2.
Another trick is to solve it with the RHS=0 first (power supply shorted out). Then solve it with the power supply turned on. You can then add these two answers together to form another solution. The two seperate solutions are known as the natural response and the forced responce respectively (and may have different frequencies).