# Math Help - First Order, Quasi-Linear, Partial Differential Equations

1. ## First Order, Quasi-Linear, Partial Differential Equations

a) Find the general solution of the equation

$(x^2 - y^2 - u^2)u_x + 2xyu_y = 2xu$

b) Find the particular solution when

$u(x,1)=x$.

2. Originally Posted by Creebe
a) Find the general solution of the equation

$(x^2 - y^2 - u^2)u_x + 2xyu_y = 2xu$

b) Find the particular solution when

$u(x,1)=x$.

The characteristic system is $\begin{cases} dx/(x^2-y^2-u^2)=dy/2xy \ (1)\cr du/dy=u/y \ \ \ (2)\end{cases}$.

Equation (2) gives $u=cy$, so substituting into (1) we obtain
$dx/dy=(x/2y)-[(1+c^2)y/2x]$. Solve this (how?) to get
$x^2=2y^2(2c^*y-d)$, where $c^*=c^*(c)=-(1+c^2)/2$. It all eventually leads to $x^2/2=-(1/2)-(u^2y/2y^2x)+d$,
and use the initial conditions $\{y=1,u=x\}$ to determine $d=d(x)$.

Oh and check the calculations...