Section Interrelated functions
[provisional cross-reference: WORK-IN-PROGRESS]In differential equations, we will encounter functions involving subscripts that are interrelated. Letβs look at an example like that.
Example: 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 \(y_2.\)
Answer.
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 donβt yet have the value of \(x_1.\) So weβll 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*}
-
Find \(y_2\) given the following information about \(x_n\) and \(y_n.\)\begin{align*} x_{n+1} \amp = x_n + 1, \amp x_0 \amp = 2\\ y_{n+1} \amp = 3\cdot x_{n+1} + y_n \amp y_0 \amp = -3 \end{align*}
Solution.
Answer.
\begin{equation*} y_2 = 18 \end{equation*} -
Find \(y_2\) given the following information about \(x_n\) and \(y_n.\)\begin{align*} x_{n+1} \amp = x_n + 1, \amp x_0 = 0\\ y_{n+1} \amp = [x_n+y_n]^2\cdot[x_{n+1} - x_n] + y_n \amp y_0 = 2 \end{align*}
Solution.
\begin{align*} n = 0: x_1 \amp = x_0 +1\\ \amp = 0 +1\\ \amp = 1\\ \amp\\ y_1 \amp = [x_0+y_0]^2\cdot[x_1 - x_0] + y_0\\ \amp = [0 + 2]^2\cdot[1 - 0] + 2\\ \amp = 6\\ \amp\\ n = 1: x_2 \amp = x_1 + 1\\ \amp = 1 +1\\ \amp = 2\\ \amp\\ y_2 \amp = [x_1+y_1]^2\cdot[x_2 - x_1] + y_1\\ \amp = [1+6]^2\cdot[2 - 1] + 6\\ \amp = 55 \end{align*}Answer.
\begin{equation*} y_2 = 55 \end{equation*} -
Find \(y_2\) given the following information about \(x_n\) and \(y_n.\)\begin{align*} x_{n+1} \amp = x_n + 2, \amp x_0 = -1\\ y_{n+1} \amp = [3x_n - y_n^2]\cdot 2 + y_n \amp y_0 = -6 \end{align*}
Solution.
\begin{align*} n = 0: x_1 \amp = x_0 +2\\ \amp = -1 +2\\ \amp = 1\\ \amp\\ y_1 \amp = [3x_0 - y_0^2]\cdot 2 + y_0\\ \amp = [3\cdot (-1) - (-6)^2]\cdot 2 + (-6)\\ \amp = [-3-36]\cdot 2 - 6\\ \amp = (-39)\cdot 2 - 6\\ \amp = -78 - 6\\ \amp = -84\\ \amp\\ n = 1: x_2 \amp = x_1 + 2\\ \amp = 1 +2\\ \amp = 3\\ \amp\\ y_2 \amp = [3x_1 - y_1^2]\cdot 2 + y_1\\ \amp = [3\cdot1 - (-84)^2]\cdot 2 + (-84)\\ \amp = [3 - 7056]\cdot 2 - 84\\ \amp = (-7053)\cdot 2 - 84\\ \amp = -14106 - 84\\ \amp = -14190 \end{align*}Answer.
\begin{equation*} y_2 = -14190 \end{equation*}
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