Hello!

Currently,I meet the following differential equation:

${f''}^4+f^4=2f'^4$

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?

thanks.

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 $sin,cos,ln,\cdots$.

existmany other real-valued solutions, but perhaps you are actually asking whether there are any otherexplicitsolutions. 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, thegeneralsolution may not be expressible in terms of elementary functions. – Robert Bryant Dec 17 '12 at 13:38neverelementary functions. – Robert Bryant Dec 17 '12 at 22:10