Suppose a $C^1$ <classical solution> existed and set $w=u+iv$. The system is precisely
$$
u_x=v_t,
\qquad
u_t=-v_x,
$$
which are the <Cauchy-Riemann equations> for $w$ as a function of the <complex number> $z=x+it$. Hence $w$ is <holomorphic function>[holomorphic] near $z=0$. Every holomorphic function is <real analytic function>[real analytic], so its restriction
$$
w(x)=w(0,x)=f(x)+ig(x)
$$
to the real axis is real analytic near zero. Its <real part> and <imaginary part> show that both $f$ and $g$ must be real analytic, contradicting the hypothesis. Therefore no such $C^1$ solution exists.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 105 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-105/1/b/solution