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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.6.117
Chapter 7, Problem 7.6.117

Solve the initial value problems in Exercises 115–120.
117. dy/dx = 1/(x√(x² - 1)), x > 1; y(2) = π

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Identify the given differential equation: \(\frac{dy}{dx} = \frac{1}{x \sqrt{x^{2} - 1}}\) with the initial condition \(y(2) = \pi\) and domain \(x > 1\).
Rewrite the differential equation in differential form: \(dy = \frac{1}{x \sqrt{x^{2} - 1}} \, dx\).
Integrate both sides with respect to \(x\): \(y = \int \frac{1}{x \sqrt{x^{2} - 1}} \, dx + C\), where \(C\) is the constant of integration.
To solve the integral \(\int \frac{1}{x \sqrt{x^{2} - 1}} \, dx\), consider using a trigonometric substitution such as \(x = \sec \theta\), which simplifies the square root expression.
After finding the antiderivative, apply the initial condition \(y(2) = \pi\) to solve for the constant \(C\), then write the explicit solution for \(y\).

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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 separable differential equation can be written as dy/dx = g(x)h(y), allowing the variables y and x to be separated on opposite sides of the equation. This enables integration with respect to each variable independently, facilitating the solution of the differential equation.
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Integration Techniques Involving Inverse Trigonometric Functions

Integrals involving expressions like 1/(x√(x² - 1)) often lead to inverse trigonometric functions such as arcsec or arccos. Recognizing these forms and applying appropriate substitution or standard integral formulas is essential to solve the integral correctly.
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Initial Value Problems (IVP)

An initial value problem specifies a differential equation along with a condition y(x₀) = y₀. Solving an IVP involves finding the general solution and then using the initial condition to determine the particular constant, yielding a unique solution curve.
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