PDE10

$u_x + 2u_y = 0,\,\,\,u(0,y) = 3\,e^{-2y}\,$

Make the seperation of variables $u(x,y) = X(x)Y(y)\,$ .

Plug back into the DE.

$X'Y+2XY'=0\,$

$\frac{X'}{X} = \frac{-2Y'}{Y} = \lambda\,$

This gives two ODES:

$X'+\lambda X=0\,$

$X(x) = c_1 \,e^{\lambda x}\,$

$-2Y' + \lambda Y = 0\,$

$Y(y) = c_2 \,e^{\frac{\lambda}{2}y}\,$

Let $A = c_1c_2\,$ .

$u(x,y) = A\,e^{\lambda(\frac{y}{2}-x)}\,$

Applying the boundary condition,

$u(0,y) = A\,e^{\lambda\frac{y}{2}} = 3\,e^{-2y}\,$

So $A=3,\,\,\,\lambda=-4\,$

The solution is

$u(x,y) = 3\,e^{4(x-\frac{y}{2})}\,$