I am given the following equation: $ (y+u)u_x + (x+u)u_y =(x+y)u$ , when: $ u(x,2x)=3x$ .
and I want to solve it using the method of characteristics.
The equations are: $ x_t = y+u , y_t=x+u , u_t = xu+yu$ and I know that I can find the solution using ODE methods (calculating the eigenvalues of the corresponding matrix, etc... ), and the solution is given by: $ x= C_1 e^{-t} + C_3 e^{2t} , y=C_2e^{-t} + C_3 e^{2t} , u= -(C_1 +C_2 ) e^{-t} +C_3e^{2t}$ . When I substitute $ x=s , y=2s, u=3s$ , I get: $ C_3 =2s , C_2 = 0 , C_1 =-s $ and everything looks great ...
The problem is that I can't figure out how to write the solution $u$ explicitly, after substituting the values for the constants $C_i $ ... In addition, when I take the initial condition to be $u(x^2 , -x^2 )= x^2 $ , I get that $u=x $ is a solution, but this solution does not solve the equation itself!
Can someone please explain to me what am I doing wrong ? (The transversality condition seems to be ok in each of the conditions above)
Thanks in advance !
