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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 9.3.21
Chapter 9, Problem 9.3.21

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = yeᵗ, y(0) = −1

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First, identify if the differential equation is separable. The given equation is \(y'(t) = y e^{t}\). Since the right side can be written as a product of a function of \(y\) and a function of \(t\), it is separable.
Rewrite the differential equation using Leibniz notation: \(\frac{dy}{dt} = y e^{t}\). Then separate variables by dividing both sides by \(y\) and multiplying both sides by \(dt\): \(\frac{1}{y} dy = e^{t} dt\).
Integrate both sides: \(\int \frac{1}{y} dy = \int e^{t} dt\). This will give you the antiderivatives on each side.
After integrating, you will have \(\ln|y| = e^{t} + C\), where \(C\) is the constant of integration. Solve for \(y\) by exponentiating both sides: \(y = \pm e^{e^{t} + C} = A e^{e^{t}}\), where \(A = \pm e^{C}\).
Use the initial condition \(y(0) = -1\) to find the constant \(A\). Substitute \(t=0\) and \(y=-1\) into the solution and solve for \(A\). Then write the particular solution.

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Key Concepts

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

Separable Differential Equations

A differential equation is separable if it can be written as a product of a function of y and a function of t, allowing the variables to be separated on opposite sides of the equation. This form enables integration with respect to each variable independently to find the solution.
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Solving Separable Differential Equations

Initial Value Problems (IVP)

An initial value problem involves solving a differential equation with a given initial condition, such as y(t₀) = y₀. The initial condition helps determine the specific solution curve among the family of solutions by fixing the constant of integration.
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Integration Techniques for Solving ODEs

Once variables are separated, integration is used to solve each side with respect to its variable. This often involves integrating exponential functions or polynomials, and applying the initial condition to solve for the integration constant to find the explicit solution.
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Integration by Parts for Definite Integrals
Related Practice
Textbook Question

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

p'(x) = 2/(x² + x), p(1) = 0

Textbook Question

45–46. Harvesting problems Consider the harvesting problem in Example 6.

If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

Textbook Question

11–16. Initial value problems Solve the following initial value problems.


y'(t) − 3y = 12, y(1) = 4

Textbook Question

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

Textbook Question

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

Textbook Question

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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