🗝️ Key Takeaways ...

• For first-order differential equations of the form

  \begin{equation*} \frac{dy}{dt} = f(t,y)\text{,} \end{equation*}

  \(f(t,y)\) acts as a slope formula for \(y(t)\) at any point \((t,y)\text{.}\)

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• Slope fields are visual tools that plot the slopes found by \(f(t,y)\) to show the flow that any solution curve must follow.
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• First-order autonomous equations have the form

  \begin{equation*} \frac{dy}{dt} = f(y) \end{equation*}

  and their solutions have the following properties:

  • Since the slopes of the solutions don’t depend on \(t\text{,}\) the slope field forms horizontal “stripes” of parallel line segments.
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  • Solutions show horizontal shift symmetry: if \(y(t)\) is a solution, so is \(y(t+c)\) for any constant \(c\text{.}\)
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• Autonomous equations also have equilibrium solutions of the form \(y(t) = c\) where \(f(c) = 0\text{.}\) These appear as rows of horizontal segments in slope fields and can be found by solving \(f(y) = 0\) for \(y\text{.}\)
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• Using phase lines or the linearization method, equilibria are classified as:

  • stable (sinks) that pull solutions in,
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  • unstable (source) that push them out, or
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  • semi-stable that do one of each.
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• Parameters can change equilibria and their stability.
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• Bifurcation diagrams show where those changes occur and summarize the system’s response to parameter variation.
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