I am trying to solve the partial differential equation
$x\ u_ x – u\ u_y = y$ with the initial condition
$u(1,y) = y$ , using the mathod of characteristics.
My problem is with y and z , I mean
$$\frac{dy}{dt} = -z$$
$$\frac{dz}{dt} = y$$
How can I solve these two?
Is the solution unique? What is the maximal domain where it is defined?
Best Answer
You have a linear system for $x,y,z$ That splits into two subsystems. You can solve the system for $(y,z)$ using matrix exponential methods or just taking the t-derivative in one of the equations and combining with the other. Either way your solution is $$ y=C_1\cos t+C_2\sin t,\quad z=C_1\sin t-C_2\cos t $$ You also have $x=C_0e^t$ so you can find the constants from your initial data on the initial curve.
The solution of a linear system is defined everywhere and unique through any given point.