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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 9.1.21
Capitolo 9, Problema 9.1.21

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
y'(t) = 3 + e⁻²ᵗ

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Recognize that the given differential equation is a first-order ordinary differential equation of the form \(y'(t) = f(t)\), where \(f(t) = 3 + e^{-2t}\).
To find the general solution, integrate both sides with respect to \(t\): \(\int y'(t) \, dt = \int (3 + e^{-2t}) \, dt\).
Split the integral on the right side into two separate integrals: \(\int 3 \, dt + \int e^{-2t} \, dt\).
Integrate each term: the integral of \(3\) with respect to \(t\) is \$3t$, and the integral of $e^{-2t}$ with respect to $t$ is \(-\frac{1}{2} e^{-2t}\) (using substitution or recognizing the exponential integral).
Combine the results and add the arbitrary constant of integration \(C\) to write the general solution: \(y(t) = 3t - \frac{1}{2} e^{-2t} + C\).

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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.

yy'(x) = 2x/(2 + y)², y(1) = −1

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

y(t) = sec² t/(2y), y(π/4) = 1

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12–16. Sketching direction fields Use the window [-2, 2] x [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed.

y'(t) = 4−y, y(0) = −1

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