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Section Autonomous Equations
When the first-order differential equation
(22) contains no explicit
\(t\) terms, it reduces to:
\begin{equation}
\frac{dy}{dt} = f(y).\tag{23}
\end{equation}
This is called an autonomous differential equation .
โAutonomousโ means โself-governing.โ In these equations, the rate of change of
\(y\) depends only on
\(y\) itself, not on time
\(t\text{.}\) The systemโs behavior is determined entirely by its current state. Think of a spring: it pushes back the same way no matter the time of day. Only how far itโs compressed matters, not what time it is.
Subsection Slope Fields of Autonomous Equations
Autonomous equations have a distinctive look in their slope fields. Consider:
\begin{equation*}
y' = y^2 - 1\text{.}
\end{equation*}
Here, the slope at any point
\((t,y)\) depends only on
\(y\text{.}\) Moving up or down (changing
\(y\) ) changes the slope, but sliding left or right (changing
\(t\) ) does not. The result is a โstripedโ slope fieldโeach horizontal line has the same slope pattern all the way across.
Figureย 94.(a) illustrates this. As you go up the plane, the slope segments gradually rotate, reflecting how
\(f(y)\) changes with
\(y\text{.}\) But moving sideways leaves the segments fixedโthe slopes donโt shift with
\(t\text{.}\)
(a) Rotating slope segments for changing \(y\) and fixed slope segments for changing \(t\) (b) Horizontal shift symmetry of solutions to autonomous differential equations Figure 94. Symmetry in the slope field for \(y' = y^2 - 1\) This symmetry isnโt just in the slope field, it shows up in the
solutions themselves. As seen in
Figureย 94.(b) , if you know one solution curve for an autonomous equation, you can create others simply by shifting that solution horizontally. Thatโs because the equation doesnโt โknowโ what time it is; it only cares about
\(y\text{.}\)
๐ค Wrap-Up.
๐๏ธ Key Takeaways...
A first-order autonomous equation has the form
\begin{equation*}
\frac{dy}{dt} = f(y).
\end{equation*}
Its slope field forms horizontal โstripesโ: the slope is constant along each horizontal line because it depends only on \(y\text{.}\)
Solutions show horizontal shift symmetry: if \(y(t)\) is a solution, so is \(y(t+c)\) for any constant \(c\text{.}\)
Check Your Understanding.
Checkpoint 95 . ๐โ Autonomous Equations.
(a) ๐โ What does a slope field represent?
Suppose you compute the slope of an autonomous differential equation be
\(3\) at the point
\((2, 1)\text{.}\) What is the slope at
\((-3, 1)\text{?}\)
\(0\) This is incorrect. The slope depends only on \(y\text{,}\) not on \(t\text{.}\)
\(-3\) This is incorrect. The slope function is \(f(y)\text{,}\) so \(t\) doesnโt affect the result.
\(1\) No, remember that the slope at a point \((t, y)\) depends solely on \(y\text{.}\)
\(3\) Correct. Since \(y = 1\) at both points, the slope is the same: \(f(1) = 3\text{,}\) regardless of \(t\text{.}\)
Impossible to answer.
This is incorrect. The value of \(f(y)\) is determined entirely by \(y\text{,}\) so this is answerable.
(b) ๐โ Shifting Solutions.
Suppose
\(y(t)\) is a solution to the autonomous equation
\(y' = f(y)\text{.}\) Which of the following must also be a solution?
\(\ y(t + 3)\) Exactlyโautonomous equations ignore the clock. Shifting in time just slides the solution along the \(t\) -axis.
\(\ y(t) + 3\) Adding to \(y\) changes the function itselfโthis doesnโt preserve the solution.
\(\ 3\ y(t)\) Scaling \(y\) is not guaranteed to produce another solution unless the DE is linear, which this one may not be.
\(\ y(-t)\) Flipping time is not generally a symmetryโit changes how \(y\) evolves.
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