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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.1.27
Capitolo 9, Problema 9.1.27

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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Recognize that the given differential equation is a second-order ordinary differential equation of the form \(u''(x) = f(x)\), where \(f(x) = 55x^{9} + 36x^{7} - 21x^{5} + 10x^{-3}\).
To find the general solution \(u(x)\), integrate the right-hand side function \(f(x)\) twice with respect to \(x\). The first integration will give \(u'(x)\), and the second integration will give \(u(x)\).
Perform the first integration: calculate \(u'(x) = \int (55x^{9} + 36x^{7} - 21x^{5} + 10x^{-3}) \, dx\). Integrate each term separately using the power rule for integration, which states \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n \neq -1\).
After finding \(u'(x)\), perform the second integration: calculate \(u(x) = \int u'(x) \, dx\). Again, integrate each term separately and add a new arbitrary constant of integration.
Combine the results to write the general solution as $u(x) = $ (the expression from the second integration) \(+ C_1 x + C_2\), where \(C_1\) and \(C_2\) are arbitrary constants representing the general solution's family.

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Second-Order Differential Equations

A second-order differential equation involves the second derivative of an unknown function. Solving such equations means finding a function whose second derivative satisfies the given equation. The general solution includes all possible functions that fit the equation, often expressed with arbitrary constants.
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Integration to Find General Solutions

To solve u''(x) = f(x), integrate the right-hand side twice with respect to x. Each integration introduces an arbitrary constant, reflecting the family of solutions. This process transforms the differential equation into an explicit formula for u(x).
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Handling Polynomial and Negative Powers in Integration

When integrating terms like x⁹ or x⁻³, apply the power rule: ∫x^n dx = x^(n+1)/(n+1) for n ≠ -1. For negative powers, ensure the integral is defined and carefully add constants. This technique is essential for integrating the given right-hand side accurately.
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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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17–20. Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it.

y(t) = 8t⁶ - 3; ty'(t) - 6y(t) = 18, y(1) = 5

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Orthogonal trajectories Use the method in Exercise 44 to find the orthogonal trajectories for the family of circles x² + y² = a²

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

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