PDEMOC3

$y^{-1} u_x + u_y = 0, u(x,1)=x^2\,$

Let $x = x(y)\,$ so $u = u(x(y),y)\,$ and let $x(1) = x_1\,$ .

Now $u_y = u_x x'(y) + u_y\,$ .

Let $u_y = u_x x'(y) + u_y = y^{-1} u_x + u_y = 0\,$ so $u\,$ is constant with respect to $y\,$ and $u(x(y),y) = u(x(1),1) = x_1^2\,$ .

$x'(y) = y^{-1}\,$

$x(y) = \ln y + x_1\,$

The solution is:

$u(x,y) = (x-\ln y)^2\,$