I have spent some time thinking about the trigonometric identities $\cos^2 x+\sin^2 x=1$ and $\cosh^2 x-\sinh^2 x=1$. We use these functions to solve integrals with the terms $\sqrt{1-x^2}$ and $\sqrt{1+x^2}$ via a simple change of variable, using the derivative relationships between these functions.
I was wondering if there is a way to define new functions $f$ and $g$ that satisfy $[f(x)]^4+[g(x)]^4=1$, as well as analogous derivative relationships like $f'=g$ and $g'=-f$. If such functions could be defined, then I would imagine that solving integrals with terms like $\sqrt{1-x^4}$, etc. could be done in terms of these functions.
Attempting to do so gives me nonlinear ODE's like $$(f')^4=1-f^4.$$ I was wondering if this has already been done, and if not, is what I am attempting to do reasonable? Can I define these function via these ODE's? I would appreciate references.
Edit: As was pointed out by Oscar Lanzi, the equations $f'=g$ and $g'=-f$ are inconsistent with the desired $f^4+g^4=1$. However, any "nice" choice of differential equations is ok (as I would like to make a simple change in variables in an integral); perhaps there is a different system of equations that are consistent with $f^4+g^4=1$?
You can't have $f'=g,g'=f$ and also $f^4+g^4=1$. For the differential equations imply
$f''=-f$ — eliminate $g$
$g(df/dg)=-f$ — reintroduce $g=f'$ for the autonomous equation
$f^{\color{blue}{2}}+g^{\color{blue}{2}}=C$ — separate and integrate
You're stuck with the exponent $2$.