I am interested in getting a good bound for solution of the following ODE:
$$ f''(t) + n^2 f(t) = (\sin(\theta t))^n$$
with the boundary condition $f(0) = f'(0) = 0$ and $t \in [0,1]$, where $\theta$ might not be an integer multiple of $\pi$ (in fact you can take $\theta < \pi/2$ so that the left hand side is almost flat at $0$ with monotone derivatives etc) . Numerical computations show that the solution is bounded by a constant independent of $n$. But I have very little experience with ODEs to prove anything about it. The only thing I did was Fourier transform, but there would be terms of the form $f(1) - f(0)$ and $f'(1) - f'(0)$ in the estimate of the Fourier coefficients of $f$ which I don't know how to deal with.


I am also interested in a good source of lecture notes where bounds for ODE solutions are discussed. The question came up in analyzing a random walk on the special orthogonal group. 



bounds on solution of an ODE
Multiplying by $f{\ }'$ you get
$$(f{\ }'^2+n^2 f{\ }^2)'=2\sin(\theta t)^n f{\ }'$$
so that the quantity $E(t)=f{\ }'^2+n^2 f^2$ satisfies the inequality
$$E'(t)\le 2 \sqrt{E(t)} \qquad \qquad(1)$$
i.e.
$$\sqrt{E(t)}'\le1$$
i.e.
$$\sqrt{E(t)}\le t$$
since $E(0)=0$. You can improve the factor 2 in eq. (1) of course since $\theta$ is small, and get sharper estimates.


