# Solving the Cauchy problem $u(0,y) = \sin y$ for PDE $u_x = yu_y$

How can I solve the Cauchy Problem $\frac{\partial u}{\partial x}=y\frac{\partial u}{\partial y}$ with the boundary condition $u(0,y)=\sin y$?

I can't figure out how to go about solving this PDE or even what the characteristic ODE will be. Can someone please provide a solution?

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{dy}{dt}=-y$ , letting $y(0)=y_0$ , we have $y=y_0e^{-t}=y_0e^{-x}$

$\dfrac{du}{dt}=0$ , letting $u(0)=f(y_0)$ , we have $u(x,y)=f(y_0)=f(e^xy)$

$u(0,y)=\sin y$ :

$f(y)=\sin y$

$\therefore u(x,y)=\sin e^xy$