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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 9.1.36
Chapter 9, Problem 9.1.36

33–42. Solving initial value problems Solve the following initial value problems.
y'(x) = 4 sec² 2x, y(0) = 8

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Identify the given differential equation and initial condition: \(y'(x) = 4 \sec^{2}(2x)\) with \(y(0) = 8\).
Recall that to solve the initial value problem, we need to find the antiderivative (integral) of \(y'(x)\) to get \(y(x)\).
Set up the integral: \(y(x) = \int 4 \sec^{2}(2x) \, dx + C\), where \(C\) is the constant of integration.
Use a substitution to integrate: let \(u = 2x\), so \(du = 2 \, dx\) or \(dx = \frac{du}{2}\). Rewrite the integral in terms of \(u\).
Integrate \(4 \sec^{2}(2x) \, dx\) using the substitution, then apply the initial condition \(y(0) = 8\) to solve for \(C\).

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Key Concepts

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

Initial Value Problems (IVPs)

An initial value problem involves a differential equation along with a specified value of the unknown function at a particular point. Solving an IVP means finding a function that satisfies both the differential equation and the initial condition, ensuring a unique solution.
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Initial Value Problems

Integration of Trigonometric Functions

Solving the given differential equation requires integrating the derivative function, which involves trigonometric functions like sec²(2x). Recognizing that the integral of sec²(u) du is tan(u) is essential, along with applying substitution when the argument is a function of x.
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Introduction to Trigonometric Functions

Applying Initial Conditions to Determine Constants

After integrating, the solution includes an arbitrary constant. Using the initial condition y(0) = 8 allows us to substitute x = 0 and y = 8 into the general solution to solve for this constant, yielding the particular solution to the IVP.
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Initial Value Problems Example 1
Related Practice
Textbook Question

25–28. Two steps of Euler’s method For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.


y′(t) = 2−y, y(0) = 1; Δt = 0.1

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Textbook Question

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(x) = sin y, y(−2) = 1/2

Textbook Question

The general solution of a first-order linear differential equation is y(t) = Ce⁻¹⁰ᵗ − 13. What solution satisfies the initial condition y(0) = 4?

Textbook Question

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


y'(x) = −y + 2

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Textbook Question

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P(1−P/800); P(0) = 100, P(0) = 400, P(0) = 700

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) = t lnt + 1

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