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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 9.3.9
Capitolo 9, Problema 9.3.9

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.
y'(t) = eʸᐟ²sin t

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Rewrite the given differential equation in terms of \( y \) and \( t \): \( \frac{dy}{dt} = e^{y/2} \sin t \).
Separate the variables by bringing all terms involving \( y \) to one side and all terms involving \( t \) to the other side: \( e^{-y/2} dy = \sin t \, dt \).
Integrate both sides: \( \int e^{-y/2} \, dy = \int \sin t \, dt \).
Evaluate the integrals: for the left side, use substitution to integrate \( e^{-y/2} \), and for the right side, recall that \( \int \sin t \, dt = -\cos t + C \).
After integration, solve the resulting equation explicitly for \( y \) as a function of \( t \), including the constant of integration.

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Separable Differential Equations

A separable differential equation can be written as a product of a function of the dependent variable and a function of the independent variable. This allows the variables to be separated on opposite sides of the equation, enabling integration with respect to each variable independently.
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Solving Separable Differential Equations

Integration Techniques

Solving separable equations requires integrating both sides after separation. Familiarity with integrating exponential functions and trigonometric functions, such as sin(t), is essential to find the antiderivatives and express the solution explicitly.
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06:18
Integration by Parts for Definite Integrals

Implicit and Explicit Solutions

After integration, solutions may be implicit or explicit. An explicit solution expresses the dependent variable directly as a function of the independent variable, which often involves algebraic manipulation or applying inverse functions to isolate the dependent variable.
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Finding The Implicit Derivative
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Domanda del libro di testo

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

y'(t) = cos² y, y(1) = π/4

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What is a separable first-order differential equation?

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33–42. Solving initial value problems Solve the following initial value problems.

y'(t) = 1 + eᵗ, y(0) = 4

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9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.


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23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.


B′(t) = 0.005B − 500, B(0) = 50,000

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15–16. {Use of Tech} Solving logistic equations Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and P₀ the initial population.


r=0.2, K=300, P₀=50

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