Consider the first order equation:

\begin{equation}

u_t + t x u_x = 0.

\label{eq-1}

\end{equation}

**a.** Find the characteristic curves and sketch them in the $(x,t)$ plane.

**b.** Write the general solution.

**c.** Solve equation (\ref{eq-1}) with the initial condition $u(x,0)= e^{-x^2}$. Explain why the solution is fully determined by the initial condition.

**d.** Describe domain in which solution of

\begin{equation}

u_t + t x^2 u_x = 0, \qquad x>0

\label{eq-2}

\end{equation}

is fully determined by the initial condition $u(x,0)=g(x)$ ($x>0$)?