Rewrite $u(f(x,y))=u(x)\Psi(\frac{w(x,y)}{u(x)})$ as: $\frac {u(f(x,y))}{u(x)}=\Psi(\frac {w(x,y)}{u(x)})$ Since $\Psi$ can be arbitrary, we can further rewrite it as: $\Phi(\frac{u(f(x,y))}{u(x)})=\Gamma(\frac {w(x,y)}{u(x)})$ where $\Psi=\Phi^{-1}(\Gamma)$ Above equation shows that any $w(x,y),u(x)$ which makes $(\frac{w(x,y)}{u(x)})$ span into $(\Phi(\frac{f(x,y)}{u(x)})$ (\Phi(\frac{u(f(x,y))}{u(x)})$is a solution. For example, given$f(x,y)=ax+by$, select$u=x$, then:$\Phi(\frac{f(x,y)}{u(x)})= \Phi(\frac{u(f(x,y))}{u(x)})= \Phi(a+b\frac{y}{x})$which can be spanned by$\frac {y}{x}$, so$w=y$is a solution. Question: how to find the appropriate$w(x,y),u(x)$to span into$\Phi(\frac {u(f(x,y))}{u(x)})$4 added 3 characters in body So far what I can think of on this question is: Rewrite$u(f(x,y))=u(x)\Psi(\frac{w(x,y)}{u(x)})$as:$\frac {u(f(x,y))}{u(x)}=\Psi(\frac {w(x,y)}{u(x)})$Since$\Psi$can be arbitrary, we can further rewrite it as:$\Phi(\frac{u(f(x,y))}{u(x)})=\Gamma(\frac {w(x,y)}{u(x)})$where$\Psi=\Phi^{-1}(\Gamma)$Above equation shows that any$w(x,y),u(x)$which makes$(\frac{w(x,y)}{u(x)}$(\frac{w(x,y)}{u(x)})$ span into $(\Phi(frac{f(x,y)}{u(x)})$ (\Phi(\frac{f(x,y)}{u(x)})$is a solution. For example, given$f(x,y)=ax+by$, select$u=x$, then:$\Phi(frac{f(x,y)}{u(x)})= \Phi(\frac{f(x,y)}{u(x)})= \Phi(a+b\frac{y}{x})$which can be spanned by$\frac {y}{x}$, so$w=y$is a solution. Question: how to find the appropriate$w(x,y),u(x)$to span into$\Phi(\frac {u(f(x,y))}{u(x)})$3 added 18 characters in body; deleted 6 characters in body; deleted 7 characters in body So far what I can think of on this question is: Rewrite$u(f(x,y))=u(x) \Psi(\frac{w(x,y)}{u(x)})$u(f(x,y))=u(x)\Psi(\frac{w(x,y)}{u(x)})$ as: $\frac {u(f(x,y))}{u(x)}=\Psi(\frac {w(x,y)}{u(x)})$ Since $\Psi$ can be arbitrary, we can further rewrite it as: $\Phi(\frac{u(f(x,y))}{u(x)})=\Gamma(\frac {w(x,y)}{u(x)})$ where $\Psi=\Phi^{-1}(\Gamma)$ Above equation shows that any $w(x,y),u(x)$ which makes $(\frac{w(x,y)}{u(x)}$ span into $(\Phi(frac{f(x,y)}{u(x)})$ is a solution.
For example, given $f(x,y)=ax+by$, select $u=x$, then: $\Phi(frac{f(x,y)}{u(x)})= \Phi(a+b\frac{y}{x})$ which can be spanned by $\frac {y}{x}$, so $w=y$ is a solution.
Question: how to find the appropriate $w(x,y),u(x)$ to span into $\Phi(\frac {u(f(x,y))}{u(x)})$