1. Sine wave decomposition.

Suppose you are given a function on an interval, $f(x): [0, 1] \to \R$, such that $f(x)$ vanishes on both end points $f(0)=f(1)=0$. Such function $f(x)$ can be expressed as a sum of 'sine waves'. $f(x) = \sum_{n=1}^\infty c_n \sin(n \pi x).$

Can you figure out a way to determine the coefficients $c_n$?

2. Consider the following equation, for $t>0$, $f'(t) + f(t) = 0$ And suppose $f(0) = 1$. Can you solve $f(t)$ for $t > 0$?

3. Consider the following equation, for $t>0$, $(d/dt + 1) (d/dt + 2) f(t) = 0$ And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?

4. Consider the following equation, for $t>0$, $(d/dt + 1) (d/dt + 1) f(t) = 0$ And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?

5. Consider the following equation, for $t>0$, $[(d/dt)^2 + 1] f(t) = 0$ And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?