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Consider the ordinary dierential equation $y''(t) + py'(t) +\frac{p^2}{4}y(t) = 0$ on $\mathbb{R}$ with $p \in \mathbb{C}$ purely imaginary and $p \ne 0$. Note that solutions will be complex valued, not necessarily real valued.
(a) Show that the equation has at least one unbounded solution.
(b) If $f$ is any unbounded solution then show that the limit $\lim_{|t|\to \infty}|\frac{f(t)}{t}|$ exists and is nonzero.


first of all I want to say that I don't know how to solve solving a second order linear equation with complex constant function and know only the case for real variable.
If I proceed in same manner then I get $y=(A+Bt)e^{-pt/2}$ is the solutions.
The linearly independent solutions are $e^{-pt/2}$ and $te^{-pt/2}$.
here is some confusion.can I take $t$ as real number?
If so then $te^{-pt/2}$ is unbounded and the given limit is $1$. Am I right? and what should be the case if $t$ is not real? somebody help me please.
Thanks for your time.


also please tell me few references where I can found these types of problems.thanks again.

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1 Answer 1

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You've got it. You've stated that the ODE holds on $\mathbb{R}$, so $t$ is always real valued. In the event you want to extend your solutions to $z \in \mathbb{C}$, you can certainly do so via $$ y(z) = (A+Bz) e^{-pz/2}.$$ Then the solution will no longer satisfy $$\lim_{|z|\to \infty} \lvert y(z)/z \rvert < \infty$$ since $p = i r$ (for some fixed real $r$) means that along the line $z = is$ (with $s \in \mathbb{R}$) the solution blows up exponentially (at least in one direction).

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