Solve $\frac{\partial u}{\partial t}+(x^2+1)\frac{\partial u}{\partial x}=0$ with initial condition $u(x,0)=f(x),x\in\mathbb R.$
Under which conditions does the solution exists?
Is it unique?
What I've done:
The general solution is $u(x,t)=\phi(t-\arctan(x))$.
Applying the initial condition we have $u(x,t)=f(\tan(t-\arctan(x)))$.
Thus the solution exists when $(t-\arctan(x)) \in I,$ where $I=\big(\frac{-(k+1)\pi}{2},\frac{(k+1)\pi}{2}\big),k\in\mathbb N.$
Am I correct in the solution ?
How do I check uniqueness?