Find functions that satisfy this equation.

Question

Give some examples of functions, $F$ and $G$ such that $$x=\sqrt{F(x)+G(x)\sqrt{F(x+n)}}-\sqrt{F(x+n)}.$$ $n$ can be a constant.

[Edit]: with $n\gt{0}$

Answer 1

Just choose some expression for $F$ and solve for $G$. $F(x)=x^2$ makes the things easy, for example: $$x=\sqrt{x^2+G(x)(x+n)}-(x+n)$$ $$(2x+n)^2=x^2+G(x)(x+n)$$ $$G(x)=\frac{3x^2+4xn+n^2}{x+n}=3x+n$$