I'll help with the first. The characteristic curve is where your PDE reduces to an ODE. So

So with or , then along

(1)

your PDE becomes

(2)

Solving (1) gives - the characteristic curves are straight lines through the origin. To see this more clearly, if we introduce new coordinates

then

and your PDE becomes which is (2)

The solution of the reduced ODE is

and the solution of the original PDE is

Then use your initial condition: when then which gives

so

which gives as your final solution