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Chapter 0

1. Find all functions $x(t),~y(t)$ satisfying $$x'(t) = y(t) - x(t)$$ $$y'(t) = 3x(t) - 3y(t)$$. Find the Particular pair of functions satisfying $x(0)=y(0)=1/2$

   Solution :

   We solve this by eliminating variables.

2. Find the function $f(n), n = 0,1,…,10$ that satisfies $$\begin{equation*} f(n) = \frac{1}{4}f(n-1)+\frac{3}{4}f(n+1), n=1,2,..,9, \end{equation*}$$ and $f(0)=0,~f(1)=1$.

   Solution :

3. The Fibonacci numbers $F_n$ are defined by $F_1=1,~F_2=1$ and for $n>2$, $F_n=F_{n-1}+F_{n-2}$. Find a formula for $F_n$ by solving the difference equation.

   Solution :

4. Find the function $f(n),~n=0,1,2,…$ that satisfies:

$$\begin{equation*} f(0) = 0,\end{equation*}$$ $$\begin{equation*} f(n) = \frac{1}{3}f(n-1)+\frac{1}{3}f(n+1)+\frac{1}{3}f(n+2),~n\geq 1,\end{equation*}$$ $$\begin{equation*} \lim_{n\rightarrow\infty} f(n) =1.\end{equation*}$$

**Solution :**

5. Find all functions $f$ from the integers to the real numbers satisfying $$\begin{equation*} f(n) = \frac{1}{2}f(n+1) + \frac{1}{2}f(n-1) -1 \end{equation*}$$

   Solution :

6. (a) Find all functions $f$ from the real numbers to the real numbers such that for all $x$, $$\begin{equation*} f''(x)+f'(x) +f(x) = 0 \end{equation*}$$

   Solution :

   (b) Find all functions $f$ from the integers to the real numbers such that for all $n$, $$\begin{equation*} f(n+2) = -f(n) -f(n+1) \end{equation*}$$

   Solution :