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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 9.1.24
Capitolo 9, Problema 9.1.24

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
p'(x) = 16/x⁹ - 5 + 14x⁶

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Identify the given differential equation: \(p'(x) = \frac{16}{x^{9}} - 5 + 14x^{6}\). This means the derivative of \(p(x)\) with respect to \(x\) is given by this expression.
To find the general solution \(p(x)\), integrate both sides of the equation with respect to \(x\): \(p(x) = \int \left( \frac{16}{x^{9}} - 5 + 14x^{6} \right) \, dx\).
Rewrite the integrand to make integration easier: \(\frac{16}{x^{9}} = 16x^{-9}\), so the integral becomes \(\int \left( 16x^{-9} - 5 + 14x^{6} \right) \, dx\).
Integrate each term separately using the power rule for integration: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
After integrating each term, combine the results and add the arbitrary constant of integration \(C\) to express the general solution \(p(x)\).

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General Solution of a Differential Equation

The general solution of a differential equation includes all possible solutions and contains arbitrary constants representing the family of curves satisfying the equation. For first-order equations, integrating the derivative function yields the general solution plus a constant of integration.
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Solutions to Basic Differential Equations

Integration of Functions

Integration is the reverse process of differentiation and is used to find the original function from its derivative. To solve p'(x), integrate each term separately with respect to x, applying power rule integration and adding an arbitrary constant.
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Integrals of General Exponential Functions

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like 16/x⁹ and 14x⁶ by rewriting negative exponents and applying the formula correctly.
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Power Rule for Indefinite Integrals
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33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.

z(x) = (z² + 4)/(x² + 16), z(4) = 2

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17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y'(t) = eᵗʸ, y(0) = 1

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Explain why the graph of the solution to the initial value problem y'(t) = t²/(1 - t), y(-1) = ln 2 cannot cross the line t = 1.

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5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

(t² + 1)³yy'(t) = t(y² + 4)

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33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.

y'(t) = 2t²/(y² − 1), y(0) = 0

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Domanda del libro di testo

Case 2 of the general solution Solve the equation y′(t) = ky + b in the case that ky + b < 0 and verify that the general solution is y(t) = Ceᵏᵗ − b/k.

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