🌌 Example 6. Identify the Variables.
Identify the dependent and independent variables in each equation:
\begin{equation*}
\frac{dP}{ds} + \frac{P}{s^2} = 17s,\qquad
u'' + t^2 u = 0,\qquad
Q'' = 11Q
\end{equation*}
Solution.
The first equation contains \(P\) and \(s\text{,}\) but the presence of \(dP/ds\) implies \(P\) changes as \(s\) changes. So,
-
\(P\) is the unknown function we solve for that depends on \(s\text{.}\)
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dependent variable \(\rightarrow P\text{,}\) independent variable \(\rightarrow s\text{.}\)
The middle equation contains \(u\) and \(t\text{,}\) but \(u''\) is the second derivative of \(u\text{.}\) So, \(u\) must change as \(t\) changes. Therefore:
-
\(u\) is the unknown function we solve for that depends on \(t\text{.}\)
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dependent variable \(\rightarrow u\text{,}\) independent variable \(\rightarrow t\text{.}\)
Only \(Q\) appears in the last equation, but \(Q''\) indicates that \(Q\) is changing, and so it must be the dependent variable. Typically, the independent variable will be clear from the context of the problem, but in this case, just assume whatever variable you like.
