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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

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.


A pot of boiling soup (100°C) is put in a cellar with a temperature of 10°C. After 30 minutes, the soup has cooled to 80°C. When will the temperature of the soup reach 30°C 

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Textbook Question

Explain how a stirred tank reaction works.

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Textbook Question

Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem

P'(t) = rP (1-P/K), P(0) = P₀

is P(t) = K/((K/P₀ − 1)e⁻ʳᵗ + 1)

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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

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

u'(x) = e²ˣ⁻ᵘ

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Textbook Question

39–42. Special equations A special class of first-order linear equations have the form a(t)y'(t)+a'(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form

a(t)y'(t) + a'(t)y(t) = d/dt (a(t)y(t)) = f(t). 

Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems. 


t³y′(t) + 3t²y = (1 + t)/t, y(1) = 6

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