Solve ux 1 yx uy 1 y u u1 y psiy where psiy is a given

Solve

u_x + 1 + y/x u_y = 1 + y + u, u(-1, y) = psi(y) where psi(y) is a given function.

Solution

1.ux+(1+y)/x.uy=1+y+x

dx/1=xdy/1+y=du/1+y+x

Take first two

dx/x=dy/1+y

Take integration , logx=log(1+y)+logc,c is constant

log(x/1+y)=logc

x/(1+y)= c

Now take addition of first two and last one

dx+dy/(1+x+y)=du/(1+x+y)

dx+dy=du

x+y-u= c\'

Hence solution is phi(x/(1+y),(x+y-u))=0,phi is arbitrary fun