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.
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 :
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 :
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 :**
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 :
(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 :