🌌 Example 364.
Suppose:
\begin{align*}
x_{n+1} \amp = x_n + 0.5,\\
y_{n+1} \amp = [y_n\cdot (x_n)^2 + 1]x_{n+1} + y_n,\\
x_0 \amp = 3, \mbox{ and}\\
y_0 \amp = 2.
\end{align*}
Find the values of \(y_1\) and \(y_2.\)
Solution.
First we would try to find \(y_1.\) Notice that when we use the formula, we have
\begin{equation*}
y_1 = [y_0\cdot (x_0)^2 + 1]x_1 + y_0.
\end{equation*}
While we do have the values of \(x_0\) and \(y_0,\) we don’t yet have the value of \(x_1.\) So we’ll find that first:
\begin{align*}
x_1 \amp = x_0 + 0.5\\
\amp = 3 + 0.5\\
\amp = 3.5
\end{align*}
Now we have all the information we need to find \(y_1:\)
\begin{align*}
y_1 \amp = [y_0\cdot (x_0)^2 + 1]x_1 + y_0\\
\amp = [2\cdot 3^2 + 1](3.5) + 2\\
\amp = 68.5
\end{align*}
Now we proceed in the same manner. First we find \(x_2:\)
\begin{align*}
x_2 \amp = x_1 + 0.5\\
\amp = 3.5 + 0.5\\
\amp = 4
\end{align*}
Then we find \(y_2:\)
\begin{align*}
y_2 \amp = [y_1\cdot (x_1)^2 + 1]x_2 + y_1\\
\amp = [68.5\cdot 3.5^2 + 1](4) + 68.5\\
\amp = 3429
\end{align*}
