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Currently,I meet the following differential equation:


we can easily find the trivial solutions:$a e^{bx}$,where $a$ is an arbitrary constant,$b=\pm i$ or $b=\pm 1$,does there exist any other real-valued solutions?


Add:many thanks to Robert Bryant,just as what has been pointed out:what I want to get is the explicit solutions which can be expressed in elementary functions,such as $sin,cos,ln,\cdots$.

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closed as off topic by Denis Serre, diverietti, Anthony Quas, Robert Bryant, Alexandre Eremenko Dec 17 '12 at 15:20

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Of course there exist many other real-valued solutions, but perhaps you are actually asking whether there are any other explicit solutions. The answer is 'yes', and one can find them by the standard symmetry methods, since the equation you write down has an obvious $2$-parameter family of symmetries. However, the general solution may not be expressible in terms of elementary functions. –  Robert Bryant Dec 17 '12 at 13:38
@Robert Bryant,many thanks for your kind help,just as what has been pointed out,I'm wondering whether there exists explicit solutions. –  gwyepsilon Dec 17 '12 at 14:23
Probably the wrong forum for the question... $x$ does not appear explicitly, so convert to first-order ODE, which happens to be homogeneous... –  Gerald Edgar Dec 17 '12 at 14:31
@gwyepsilon: Just in case you need a hint: If you consider the function $q = f/f'$ on a real solution other than $f\equiv0$, which must always satisfy $q^4 < 2$, then you'll find that, on a solution, either $q^2\equiv1$ (which are your trivial solutions) or else one has an equation of the form $$q' = 1\pm q(2{-}q^4)^{1/4},$$ and the non-constant solutions of these two equations are never elementary functions. –  Robert Bryant Dec 17 '12 at 22:10
@Robert Bryant,thanks for your kind help. –  gwyepsilon Dec 19 '12 at 15:23

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