= Solution
Both functions satisfy the constant-coefficient <homogeneous linear differential equation>
$$
\boxed{y'''+y'=0.}
$$
Its <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 $-B=0$, hence
$$
\boxed{y(x)=\sin x.}
$$
Here 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 \b[a unique solution on the entire half-line].
Back to article page
