Characteristics for 2nd order differential equations If I have an equation 
$p(x)\frac{\partial^2u}{\partial x^2} + r(x)\frac{\partial^2u}{\partial x\partial y} + q(x)\frac{\partial^2 u}{\partial y^2}=f(x,y,u)$
Where $f$ maybe contains first partial derivatives for $u$. 
Can anyone give me a worked example of how to solve this using the method of characteristics. 
Thanks in advance. 

See these lecture notes on the method of characteristics for second-order PDEs, where the method is applied to steady isentropic flow (gasdynamics)
