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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 9.3.18
Chapter 9, Problem 9.3.18

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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First, rewrite the differential equation in the form \( \frac{dy}{dt} = e^{ty} \) to clearly identify the variables involved.
Check if the equation is separable by trying to express it as a product of a function of \( t \) and a function of \( y \), i.e., \( \frac{dy}{dt} = g(t)h(y) \). In this case, observe that \( e^{ty} \) cannot be separated into a product of a function of \( t \) alone and a function of \( y \) alone.
Since the equation is not separable, the standard method of separation of variables does not apply here. Consider alternative methods such as substitution or recognizing the equation type.
If you attempt substitution, for example, let \( u = ty \), then express \( y \) and \( y' \) in terms of \( u \) and \( t \) to transform the equation into a potentially separable or solvable form.
After substitution, solve the resulting differential equation for \( u(t) \), then back-substitute to find \( y(t) \). Finally, apply the initial condition \( y(0) = 1 \) to determine the constant of integration.

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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 t and a function of y, 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 specifies the value of the unknown function at a particular point, providing a unique solution to a differential equation. Solving an IVP involves finding the general solution and then applying the initial condition to determine the constant of integration.
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Integration Techniques for Exponential Functions

Solving differential equations involving exponential functions often requires integrating expressions like e^(t*y). Recognizing when to use substitution or other integration methods is essential to handle these integrals and find explicit solutions.
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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

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

y''(t) = teᵗ, y(0) = 0, y'(0) = 1

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

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

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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Textbook Question

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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