1. The problem statement, all variables and given/known data I have a circuit with input source x(t), which contains also an inductor and a capacitor in series which I have found to be related to the output voltage y(t) (across the capacitor) like so: LC*d2y/dt2 + y(t) = x(t). I have also found its roots through the quadratic equation to be complex, and equal to +/- the imaginary unit j. The question asks me to show its homogenous solution has the form K1ej*a*t + K2ej*b*t, where a and b are the roots I found above. 2. Relevant equations Unsure. See above for my progress so far. 3. The attempt at a solution My solution has led me here, having gotten the above equation and its roots. I just do not know how to show that I can express the solution to the differential equation in the requested form. Any help would be greatly appreciated!