(a) Show that the solution in the lecture,

$$x(t) = \frac{1}{2} - e^{-t} + \frac{1}{2}e^{-2t} - u_1(t) \left( \frac{1}{2} - e^{-(t-1)} + \frac{1}{2}e^{-2(t-1)} \right),$$

is continuous at $$t=1$$.

(b) Solve

$$\ddot x + x = 1-u_{2\pi}(t)$$, with $$x(0)=0$$ and $$\dot x(0)=0$$.