Solution
= Solution
From (a), combine the three <derivative> integrals:
$$
tJ_0''+J_0'+tJ_0=\frac2\pi\int_0^{\pi/2}\bigl(t\sin^2x\cos(t\cos x)-\cos x\sin(t\cos x)\bigr)dx.
$$
The integrand is $-\frac d{dx}[\sin x\sin(t\cos x)]$. This bracket vanishes at both endpoints, so the integral is zero. Hence $\boxed{tJ_0''+J_0'+tJ_0=0}$, also at $t=0$ by the same formula.