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I was given this problem:

Solve the system $$X'=\begin{bmatrix}1&2\\-4&-3\end{bmatrix}X$$

I found the eigenvalues $\lambda_1$= -1-2i and $\lambda_2$= -1+2i. Then solving for the eigenvectors I found $V_1$=(-1-i,2) and $V_2$=(-1+i,2).

I know the general solution should be in the form: $$X(t)=c_1 e^{\lambda_1{t}} V_1 + c_2 e^{\lambda_2{t}} V_2 $$ but I don't know how to keep going after this.

Any help is very welcome. Thanks a lot!

juan deutsch
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    https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iv-first-order-systems/matrix-methods-eigenvalues-and-normal-modes/MIT18_03SCF11_s33_7text.pdf, there are also many examples on MSE, for example: https://math.stackexchange.com/questions/2665062/solve-the-following-differential-equations-2x2-with-complex-eigenvert – Moo Jun 27 '18 at 00:58
  • Thank you very much! Sorry for the repeated question! – juan deutsch Jun 27 '18 at 01:06
  • Hint: If $X$ is a solution, then so are its real and imaginary parts. Alternatively, you can develop a general solution using the Cayley-Hamilton theorem, which doesn’t require computing eigenvectors. – amd Jun 27 '18 at 01:21
  • Check out some examples here: http://tutorial.math.lamar.edu/Classes/DE/ComplexEigenvalues.aspx – Dylan Jun 27 '18 at 05:07

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