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there is something that has always bugged me in linear second order ODEs. We say that the general solution is:

[tex]y=C_1e^{r_1x}+C_2e^{r_2x}[/tex]

where r_1 and r_2 are the solutions of the characteristic polynomial.

The cases where r1, r2 are real are pretty straightforward. If they are complex, however:

[tex]r_1=a+ib , r_2=a-ib[/tex]

after algebraic manipulation we come up with the following:

[tex]y=e^{ax}[(C_1+C_2)cos(bx)+i(C_1-C_2)sin(bx)]=e^{ax}[c_1 cos(bx)+c_2 sin(bx)][/tex]

We then say, that this is the general solution of the ODE, and if c_1, c_2 are real, the solutions are also real.

This would imply that the C_1 and C_2 coefficients are considered to be complex. However, I have not seen such an assumption in any book or description so far. In fact, they are sometimes not defined at all, simply referred to as "arbitrary constants". Does this assumption cover the possibility that they may be complex numbers? Am I missing something obvious?

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# Question about linear second order ODE solution coefficients

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