I am needing to use the Variation of parameters formula to solve a second order non-homogeneous equation. I have used this before however i now have an equation with complex imaginary roots
My second order differential equation is y'' + 2y' + 2y = exp(-t)sin(t)
so i'm working with the roots to the characteristic equation λ^2 + 2λ + 2 = 0
Do i just use the formula normally or is there a different method for complex imaginary roots?
Best Answer
Whenever there's an $i$, turn it into sines and cosines. In your characteristic equation, the roots are $\lambda = -1 \pm i$, so the general solution is
$$ y_h = C_1 e^{(-1 + i) t} + C_2 e^{(-1 - i) t}$$ $$ = C_1e^{-t}e^{it} + C_2 e^{-t} e^{-it}$$ $$ = C_1e^{-t}(\cos t + i\sin t) + C_2e^{-t}(\cos t - i\sin t) $$ $$ = (C_1 + C_2)\,e^{-t}\cos t + i(C_1 - C_2)\,e^{-t}\sin t $$ $$ = A_1 e^{-t}\cos t + A_2 e^{-t}\sin t $$