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AP Calculus AB/BC · Advanced Placement (AP) · Unit 7 of 10 · Gatekeeper unit

7Unit 7: Differential Equations

Work through these 9 topics in order. Clear the gatekeeper and the rest of the course opens up.

Contents

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  1. 7.1
    7.1 Modeling situations with differential equations

    A differential equation is a sentence about a RATE.

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  2. 7.2
    7.2 Verifying solutions for differential equations

    A proposed solution is just a candidate function.

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  3. 7.3
    7.3 Sketching slope fields

    A slope field is a picture of a differential equation: at a grid of points, draw a tiny segment whose slope is what dydx\dfrac{dy}{dx} evaluates to there.

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  4. 7.4
    7.4 Reasoning using slope fields

    Given a slope field, you can describe solutions without a formula: where they rise or fall, where equilibria sit, and roughly what a particular solution through a given point looks like.

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  5. 7.5
    7.5 Approximating solutions using Euler's method (BC)

    When you cannot solve a DE exactly, march along it in small steps.

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  6. 7.6
    7.6 Finding general solutions using separation of variables

    If a DE can be rearranged so all the yy’s (with dydy) sit on one side and all the xx’s (with dxdx) on the other, you can integrate each side separately to recover yy.

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  7. 7.7
    7.7 Finding particular solutions using initial conditions

    The general solution has a free constant — a whole family of curves.

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  8. 7.8
    7.8 Exponential models with differential equations

    Whenever a quantity changes at a rate proportional to itself, it grows or decays exponentially.

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  9. 7.9
    7.9 Logistic models with differential equations (BC)

    Real populations cannot grow forever.

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