Solve the following Cauchy problem

,

subject to

, .

Attempt:

The characteristic equations are , , .

The initial conditions are , and .

The Jacobian is and hence we expect a unique solution when and . (Is this correct?)

Now solve the characteristic equations.

.

Apply initial condition to get and hence .

.

Apply initial condition to get and hence . (Is this it for the question? Why is independent of ? What have I done wrong?)

Substitute above and into characteristic equation and we get . Integrate over and we get . Apply initial condition we get and .

From expressions of and obtained above we get

.

Therefore the characteristics is . (Do I need this characteristics at all? What should I do with it?)

Is the above attempt correct?