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?

  • $\begingroup$ Just use the quadratic formula as normal. $\endgroup$ – mattos Mar 8 '15 at 13:45
  • $\begingroup$ But how would I go about integrating something with imaginary i in it? $\endgroup$ – jack Mar 8 '15 at 13:59
  • $\begingroup$ Ok, it sounds like there's some confusion so edit your post to include the entire question and we'll go through it. $\endgroup$ – mattos Mar 8 '15 at 14:01
  • $\begingroup$ Also, here's a previous question I solved using the variation of parameters (although it doesn't have complex solutions which is where I think you're having trouble). You might like to have a look through it. $\endgroup$ – mattos Mar 8 '15 at 14:03
  • $\begingroup$ thanks Mattos but yes it is purely the complex imaginary solutions which are stumping me at the moment $\endgroup$ – jack Mar 8 '15 at 14:09

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

$$ y_h = C_1 e^{(-1 + i) t} + C_2 e^{(-1 - i) t}$$ $$ = C_1e^{-t}e^{it} + C_2 e^{-t} e^{-it}$$ $$ = C_1e^{-t}(\cos t + i\sin t) + C_2e^{-t}(\cos t - i\sin t) $$ $$ = (C_1 + C_2)\,e^{-t}\cos t + i(C_1 - C_2)\,e^{-t}\sin t $$ $$ = A_1 e^{-t}\cos t + A_2 e^{-t}\sin t $$


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