Textbook Question
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.4.21
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
Verified step by step guidance1
Rewrite the given differential equation to isolate \( \frac{dy}{dx} \). Multiply both sides by \( x \) to get \( \frac{dy}{dx} = x y e^{x^{2}} + 2 x \sqrt{y} e^{x^{2}} \).
Observe the right-hand side and consider a substitution to simplify the equation. Since the terms involve \( y \) and \( \sqrt{y} \), let \( u = \sqrt{y} \), so that \( y = u^{2} \).
Express \( \frac{dy}{dx} \) in terms of \( u \) and \( \frac{du}{dx} \). Using the chain rule, \( \frac{dy}{dx} = 2 u \frac{du}{dx} \). Substitute this and \( y = u^{2} \) into the equation.
After substitution, the equation becomes \( 2 u \frac{du}{dx} = x u^{2} e^{x^{2}} + 2 x u e^{x^{2}} \). Divide both sides by \( u \) (assuming \( u \neq 0 \)) to simplify to \( 2 \frac{du}{dx} = x u e^{x^{2}} + 2 x e^{x^{2}} \).
Rewrite the equation as \( \frac{du}{dx} - \frac{x}{2} e^{x^{2}} u = x e^{x^{2}} \). This is a linear first-order differential equation in \( u \). Identify the integrating factor \( \mu(x) = e^{-\int \frac{x}{2} e^{x^{2}} dx} \) and proceed to solve for \( u(x) \).
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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 a product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This technique simplifies solving by integrating each side independently.
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Solving Separable Differential Equations
Substitution Method
Substitution involves introducing a new variable to simplify the differential equation, especially when terms like √y or y appear together. This method transforms the equation into a more manageable form, often linear or separable.
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Integration Techniques
Solving the separated or substituted differential equation requires integration. Familiarity with integrating exponential functions and powers, as well as applying integration constants, is essential to find the general solution.
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Related Practice
Textbook Question
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
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Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Textbook Question
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt
Textbook Question
7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.
a. lim(x→∞)A(t)
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