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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.2.2
Chapter 9, Problem 9.2.2

Consider the differential equation y'(t) = t² - 3y² and the solution curve that passes through the point (3, 1). What is the slope of the curve at (3, 1)?

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1
Identify the given differential equation: \(y'(t) = t^{2} - 3y^{2}\).
Recognize that \(y'(t)\) represents the slope of the solution curve at any point \((t, y)\).
Substitute the given point \((t, y) = (3, 1)\) into the differential equation to find the slope at that point.
Calculate the value of the right-hand side by plugging in \(t = 3\) and \(y = 1\): \(3^{2} - 3 \times 1^{2}\).
Simplify the expression to find the slope \(y'(3)\) at the point \((3, 1)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Differential Equation

A differential equation relates a function with its derivatives, describing how the function changes. In this problem, y'(t) represents the derivative of y with respect to t, indicating the slope of the solution curve at any point.
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Evaluating the Derivative at a Point

To find the slope of the solution curve at a specific point, substitute the given values of t and y into the differential equation. This gives the instantaneous rate of change or slope at that point.
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Initial Condition and Solution Curve

The initial condition (3, 1) specifies a point through which the solution curve passes. It helps identify the particular solution of the differential equation and allows evaluation of the slope at that exact point.
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Related Practice
Textbook Question

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P − 0.001P²; P(0) = 10, P(0) = 40, P(0) = 80

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Textbook Question

5–10. First-order linear equations Find the general solution of the following equations.


y'(x) = −y + 2

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Textbook Question

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P(1−P/800); P(0) = 100, P(0) = 400, P(0) = 700

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Textbook Question

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

u''(x) = 55x⁹ + 36x⁷ - 21x⁵ + 10x⁻³

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Textbook Question

20–22. {Use of Tech} Solving the Gompertz equation Solve the Gompertz equation in Exercise 19 with the given values of r, K, and M₀. Then graph the solution to be sure that M(0) and lim(t→∞) M(t) are correct.


r = 0.05, K = 1200, M₀ = 90

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Textbook Question

What is the equilibrium solution of the equation y'(t) = 3y − 9? Is it stable or unstable?

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