Consider the equation $x''(t) + α(t)x(t) = 0$ where $α$ is a continuous function on $−∞ < t < ∞$ which is of period $τ$. Let $x_1$ and $x_2$ be the linearly independent solutions satisfying
$x_1(t_0)=1=x'_2(t_0)$ and $x'_1(t_0)=x_2(t_0)$.
1) Show that $W[x_1, x_2](t) = 1$ for all $t$.
2) Show that there is at least one non-trivial solution $x$ of period $τ$ if, and only if, $x_1(τ ) = 1 = x'_2(τ )$, $x_2(τ ) = x'_1(τ ) = 0 .$
The first part was easy. The Wronskian of $x_1$ and $x_2$ at $t_0$ is 1. And since the coefficient of $x'$ in the ODE is $0$, by Abel's theorem, the Wronskian is identically $1$. But I have no clue how to do the second part. Assuming $x$ is a periodic solution of period $\tau$, I was only able to show that $x_1(t_0+\tau)+x'_2(t_0+\tau)=2$.
Any hint would be very much appreciated. Thank you.