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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.PE.90
Chapter 4, Problem 4.PE.90

Initial Value Problems
Solve the initial value problems in Exercises 89–92.


dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

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Rewrite the differential equation clearly: \(\frac{dy}{dx} = \left(x + \frac{1}{x}\right)^2\).
Expand the right-hand side expression: \(\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}\).
Set up the integral to find \(y\): \(y = \int \left(x^2 + 2 + \frac{1}{x^2}\right) \, dx + C\).
Integrate each term separately: \(\int x^2 \, dx\), \(\int 2 \, dx\), and \(\int \frac{1}{x^2} \, dx\).
Use the initial condition \(y(1) = 1\) to solve for the constant of integration \(C\) after finding the general solution.

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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 solving a differential equation with a given initial condition, which specifies the value of the unknown function at a particular point. This condition allows us to find a unique solution that fits both the differential equation and the initial value.
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Initial Value Problems

Separable Differential Equations

A separable differential equation can be written so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side. This allows integration of both sides separately to find the general solution.
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Solving Separable Differential Equations

Integration of Algebraic Expressions

Solving the given differential equation requires integrating algebraic expressions, such as polynomials or rational functions. Understanding how to expand, simplify, and integrate these expressions is essential to find the explicit form of the solution.
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Simplifying Trig Expressions
Related Practice
Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec² s/10 ds

Textbook Question

Applications


Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ cos³ 𝓍/2 d𝓍

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 𝓍³ (1 + 𝓍⁴ )⁻¹/⁴ d𝓍

Textbook Question

54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)

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Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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∫ ( 3√ t + 4/t² ) dt