E078 Section 5 — Differential Equations

Calculus 2 for Engineers (WGU E078), Section 5: DE basics, direction fields, Euler's method, separable equations, Newton's law of cooling, logistic growth. Built from the zyBooks Section 5 study materials; not official WGU material.

1

Differential equation (definition)

2

Solution of a differential equation

3

How do you verify a solution to a DE?

4

Order of a differential equation

5

Order of $(y')^3 + y = x$

6

General solution vs. particular solution

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Initial-value problem (IVP)

8

How many initial values does an $n$th-order IVP need?

9

Solve $y' = f(x)$, $y(x_0) = y_0$

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Velocity DE for an object under gravity only

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Velocity of an object under gravity

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Height of an object under gravity

13

Why doesn't mass matter in the falling-object DE?

14

When is a thrown object at its maximum height?

15

Direction (slope) field

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How do you find the slope at $(x_0, y_0)$ for $y' = f(x, y)$?

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Solution curve

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Equilibrium solution

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Asymptotically stable equilibrium

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Asymptotically unstable equilibrium

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Semi-stable equilibrium

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How do you classify equilibria of $y' = g(y)$ quickly?

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Equilibria of $y' = (y - 2)(y + 1)$ and their stability

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What does a squared factor like $(y - 1)^2$ in $y' = g(y)$ usually signal?

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Euler's method formulas

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Where does Euler's method come from?

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Effect of step size $h$ in Euler's method

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Euler: $y' = x + y$, $y(0) = 1$, $h = 0.1$, find $y_1$

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Common Euler mistake

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Separable differential equation

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Autonomous differential equation

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Is $y' = xy + 3x - 2y - 6$ separable?

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Is $y' = x + y$ separable?

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Separation of variables: the 5 steps

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Why check $g(y) = 0$ before separating?

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$\ln|y| = h(x) + C \implies y = ?$

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Solve $y' = ky$

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Solve $\frac{dy}{dx} = \frac{x}{y}$ with $y(0) = -2$

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Is an implicit solution acceptable?

40

Mixing tank DE

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Salt rate in (mixing tank)

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Salt rate out (mixing tank, constant volume $V$)

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Long-run salt amount in a mixing tank

44

Newton's law of cooling (DE)

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Newton's law of cooling (solution)

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How do you find $k$ in Newton's law of cooling?

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Sign of $k$ in Newton's law of cooling

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Carrying capacity $K$

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Exponential growth model

50

Logistic differential equation

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Solution of the logistic IVP

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Equilibria of the logistic equation

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Logistic: what happens if $P_0 > K$?

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Logistic: when is the population growing fastest?

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Maximum growth rate of a logistic population

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Shape of a logistic curve ($P_0 < K/2$)

57

Read $r$ and $K$ from $P' = 0.3P - 0.0006P^2$

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Logistic: useful middle step for 'when does $P$ reach a value?'

59

Threshold population model