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what is the condition on $r_1$ and $r_2$ the roots of a second order homogeneous differential equation with constant complex coefficients so that it accepts real solutions. Given: $$ay'' + by'+cy=0 \space\space\space\space\space\space a,b,c\in \Bbb C$$ $$\exists\lambda,\mu \in \Bbb C \space\space\space \forall x : \space\space Im(y)=0$$ I wrote the case in another way I don't know if it helps $$\exists\lambda,\mu\in \Bbb C \space\space\space \forall x : \space\space$$ $$e^{Re(r_1)}Re(\lambda)\sin(Im(r_1)x) + e^{Re(r_1)}Im(\lambda)\cos(Im(r_1)x) +e^{Re(r_2)}Re(\mu)\sin(Im(r_2)x) +e^{Re(r_2)}Im(\mu)\cos(Im(r_2)x) = 0$$ Since $$y=\lambda e^{r_1x}+\mu e^{r_2x} \space\space\space\space \lambda,\mu,r_1,r_2 \in \Bbb C$$ And again I'm looking for a condition on $r_1$ and $r_2$ so that if and only if it's true (satisfied) the solution $y\space$ will accept real solutions (Other than $y=0$).

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  • $\begingroup$ I'm sorry is the question clear or should I rewrite it? thanks a lot already :) $\endgroup$ Jun 21, 2020 at 13:56
  • $\begingroup$ Your question might be answered in these notes- see real, complex, and repeated roots $\endgroup$
    – Axion004
    Jun 21, 2020 at 15:35
  • $\begingroup$ Sorry, but this site doesn't discuss when the coefficients are complex and when the vector space of functions that the solution is in is defined on the field C (complex numbers) $\endgroup$ Jun 21, 2020 at 16:05
  • $\begingroup$ You also have to put conditions on $\lambda$ and $\mu$. I.e. for $\Im[y(0)] = 0$ one must require that at least $\Im(\mu) = -\Im(\lambda)$. This doesn't necessarily mean that the addends of $y(x)$ must be complex conjugates, but examining that specific constraint does give you some constraints on $r_1, r_2$, such as $\Re(r_1) = \Re(r_2)$ $\endgroup$
    – Andy Walls
    Jun 22, 2020 at 12:07
  • $\begingroup$ You're correct, I'm sorry, I had a similar opinion and I went and ask the professor to clarify the question and I edited Mine with the correct question. $\endgroup$ Jun 22, 2020 at 17:24

1 Answer 1

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For $r_n \in \Bbb C$, to maintain $\Im[y(x)] =0$

  1. the addends of $y(x)$ will be jointly either spiraling in to the origin, spiraling out from the origin, or maintaining circular paths about the origin.

  2. the addends must maintain the same radii and angular rate around the origin. The exponential makes sure that any mismatch in initial radius and hence mismatched radial growth between the 2 addends will make it impossible to keep $\Im[y(x)] =0$ for all $x$, no matter what the angular rates.

  3. The addends can follow opposite angular directions, forcing the addends to be complex conjugates, or they can follow the same angular direction, forcing the addends to be the same radius, but separated by $\pi$ radians.

So there are 2 cases to consider for the addends

Case 1: $$\mu e^{r_2 x} = \overline{\lambda e^{r_1 x}}$$ thus $$\mu = \overline{\lambda}$$ $$r_2 = \overline{r_1}$$

Case 2: $$\mu e^{r_2 x} = \lambda e^{i\pi} e^{r_1 x} =-\lambda e^{r_1 x}$$ thus $$\mu = -\lambda$$ $$r_2 = r_1$$

Since $a, b \in \Bbb C$, it is possible to have a double root for which $\Im(r) \ne 0$.

Case 2 is probably not so interesting, as it results in $y(x) = 0$.

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