Given the initial value problem $$\begin{cases} x' = 5\text{cos}^2(tx)-x^2-5 \\ x(1)=0\end{cases}$$
I am asked to prove that it has a unique solution for the given initial condition ($x(1)=0)$
I have tried to prove it seeing that $f(t,x) = 5\text{cos}^2(tx)-x^2-5$ is Lipschitz-continuous. Using the intermediate value theorem I got that $$|f(t,x)-f(t,y)|<(10t+|x+y|)|x-y|$$ However I still have not used the fact that $x(1)=0$ at any point, how does knowing that help me prove the uniqueness of solution?