Question

Solve the following equations:
$x(a+x)=y(b+y),ax+by=(x+y{)}^2$.

Answer 1

$⟹ax-by=y^2-x^2⟶\left(I\right)$

$ax+by={(x+y)}^2⟶\left(II\right)$

$⟹\left(II\right)/\left(I\right):-$
$⟹\frac{ax+by}{ax-by}=\frac{y+x}{y-x}$
Let $y/x=p$

$⟹\frac{a+bp}{a-bp}=\frac{p+1}{p-1}$ (by componendo & dividendo)
$⟹\frac{2a}{2bp}=\frac{2p}{2}$
$⟹p=\sqrt{\frac{a}{b}}$ $y=x\sqrt{\frac{a}{b}}$
In equation $\left(I\right)$
$x(a+x)=x\sqrt{\frac{a}{b}}(b+x\sqrt{\frac{a}{b}})$
$a+x=\sqrt{ab}+x\frac{a}{b}$
$\sqrt{a}(\sqrt{a}-\sqrt{b})=x\left(\frac{a-b}{b}\right)$
$x=\frac{b\sqrt{a}}{\sqrt{a}+\sqrt{b}}$
$x=\frac{a\sqrt{b}}{\sqrt{a}+\sqrt{b}}$

Also $x=0$ and $y=0$ satisfy the equation.