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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 9.3.24
Chapter 9, Problem 9.3.24

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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First, rewrite the differential equation in Leibniz notation: \(\frac{dy}{dt} = \cos^{2} y\).
Check if the equation is separable by expressing it as a product of a function of \(y\) and a function of \(t\). Here, rewrite as \(\frac{dy}{dt} = \cos^{2} y = f(y) \cdot g(t)\), where \(f(y) = \cos^{2} y\) and \(g(t) = 1\).
Since the equation is separable, separate variables by dividing both sides by \(\cos^{2} y\) and multiplying both sides by \(dt\): \(\frac{1}{\cos^{2} y} dy = dt\).
Integrate both sides: \(\int \frac{1}{\cos^{2} y} dy = \int dt\). Recall that \(\frac{1}{\cos^{2} y} = \sec^{2} y\), and the integral of \(\sec^{2} y\) with respect to \(y\) is \(\tan y\).
After integrating, apply the initial condition \(y(1) = \frac{\pi}{4}\) to solve for the constant of integration and express the solution implicitly or explicitly in terms of \(y\) and \(t\).

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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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Initial Value Problems (IVP)

An initial value problem involves solving a differential equation with a given initial condition, such as y(t₀) = y₀. This 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 Trigonometric Functions

Solving differential equations involving trigonometric functions often requires using identities and integration methods, such as rewriting cos² y using power-reduction formulas. Mastery of these techniques is essential to integrate and solve the equation explicitly.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

What is a separable first-order differential equation?

Textbook Question

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

y'(t) = 1 + eᵗ, y(0) = 4

Textbook Question

17–20. Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it.

y(t) = 8t⁶ - 3; ty'(t) - 6y(t) = 18, y(1) = 5

Textbook Question

5–10. First-order linear equations Find the general solution of the following equations.


v'(y) − v/2 = 14

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Textbook Question

Orthogonal trajectories Use the method in Exercise 44 to find the orthogonal trajectories for the family of circles x² + y² = a²

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Textbook Question

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.