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**tautology** $\displaystyle u^{\prime\prime} + u = F(t), u(0)=0, u^{\prime}(0)=0$

$\displaystyle F(t)=\left\{\begin{array}{cc}F_0t, & 0 \leq t \leq \pi\\F_0(2\pi-t), & \pi < t\leq2\pi\\0, & 2\pi<t\end{array}\right.$

Hint: treat each time interval separately, and match the solutions in the different intervals by requiring u and u' to be continuous functions of t.

I started this problem by trying to solve for the complimentary solution. The characteristic equation is r^2 + 1 =0. Meaning r = i.

Thus, I got:

$\displaystyle u(t) = c_1cos(t)+c_2sin(t)$

Using the initial conditions, I got c1 and c2 =0. I don't think this is right though...