Solutions to functional equation $f(at+x)+g(x)+h(t,bt+x) =0 $

Question

Let $a \neq 0$ and $b \neq 0$ be fixed constants with $a \neq b$. Find all twice continuously differentiable functions $f:\mathbb{R}\rightarrow \mathbb{R}$, $g:\mathbb{R}\rightarrow \mathbb{R}$ and $h:\mathbb{R}^2 \rightarrow \mathbb{R}$ such that, \begin{align*} f(at+x)+g(x)+h(t,bt+x) =0 \end{align*} and $f(0)=g(0)=h(0,0)=0$.

Answer 1

Given $f$ and $g$ twice continuously differentiable such that $f(0)=g(0)=0$ we'll try to find $h$. Let $u=t$ and $v=bt+x$ then $x=v-bu$ and $at+x=bt+x+(a-b)t=v+(a-b)u$ So that $$h(u,v)=-f\left(v+(a-b)u\right)-g(v-bu)$$