4. a) State the Cauchy-Kovalevsky theorem
b) For the following initial value problem verify that the assumptions of the Cauchy-Kovalevsky theorem are satisfied and then find the terms of order ≤ 2 of the Taylor series of the solution about the origin:
$u_t = e^u u_x$, $u(0, x) = 1 - x + x^2$