Use separation $of$ variables $to$ find a solution $of$ the partial differential equation

$x\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+tany\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}=0$

$on$ xin$(0,)$ and yin$(0,\frac{}{2}),$ with boundary value $u(x,\frac{}{2})=x.$

$(4\mathrm{.}1)$

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}=4\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}$;$,u(0,y)=8e^{-3y}$