 Solution

Both functions satisfy the constant-coefficient homogeneous linear differential equationIts characteristic polynomial is $\lambda ({\lambda }^2+1)$, so the general real solution is $y=A+B\cos x+C\sin x$. The three initial conditions give $A+B=0$, $C=1$, and $-B=0$, henceHere the equation has continuous coefficients and nonzero highest-derivative coefficient everywhere. The uniqueness theorem for ordinary differential equations, applied to its three-component first-order system, gives a unique solution on the entire half-line.