Solving Mass Conservation with Characteristic System

[stanley.st]

Hello i want to solve

\frac{\partial \rho}{\partial t}=\frac{\partial v_1\rho}{\partial x_1}+\frac{\partial v_2\rho}{\partial x_2}

for v_1 = -x_2 and v_2=x_1

i obtain equation

\frac{\partial \rho}{\partial t}+x_2\frac{\partial\rho}{\partial x_1}-x_1\frac{\partial \rho}{\partial x_2}=0

Charakteristik system is

\begin{array}{rcl}t'&=&1\\x_1'&=&x_2\\x_2'&=&-x_1\end{array}

Thanks

 

[HallsofIvy]

You seem to have misunderstood the purpose of this forum. We are not here to do your work for you. What have you done on this problem yourself and what specific questions about it do you have?

 

[stanley.st]

I need to find two functions I_1, I_2 constant on charakterstics and write general solution

u(x,y,t)=\varphi(I_1,I_2)

I found one function

I_1=x_1^2+x_2^2

I don't know to find second one with t. Thx