Textbook Question
In Exercises 1–22, solve the differential equation.
dy + x(2y - e^(x-x²))dx = 0
Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 9, Problem 9.PE.8
In Exercises 1–22, solve the differential equation.
y' = (y²-1)x⁻¹
Verified step by step guidance1
Rewrite the given differential equation in a more explicit form: \(y' = \frac{dy}{dx} = \frac{y^{2} - 1}{x}\).
Recognize that this is a separable differential equation because the right side can be expressed as a product of a function of \(y\) and a function of \(x\).
Separate the variables by rewriting the equation as \(\frac{dy}{y^{2} - 1} = \frac{dx}{x}\).
Integrate both sides: integrate \(\int \frac{1}{y^{2} - 1} \, dy\) on the left and \(\int \frac{1}{x} \, dx\) on the right.
After integration, solve the resulting equation for \(y\) implicitly or explicitly, and include the constant of integration.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
9mWas this helpful?
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 y and a function of x, allowing variables to be separated on opposite sides of the equation. This form enables integration with respect to each variable independently to find the solution.
Recommended video:
06:06
Solving Separable Differential Equations
Integration Techniques
Solving separable equations requires integrating both sides after separation. Familiarity with integrating rational functions, such as partial fraction decomposition, is essential when dealing with expressions like (y² - 1) in the denominator or numerator.
Recommended video:
06:18Integration by Parts for Definite Integrals
Initial Conditions and General Solutions
After integrating, the solution typically includes an arbitrary constant representing a family of solutions. Applying initial conditions, if given, helps determine the particular solution. Understanding the difference between general and particular solutions is key.
Recommended video:
05:03Initial Value Problems
Related Practice
Textbook Question
In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.
Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.
b. How many pounds of salt are in the tank after 1 minute? after 30 minutes?
1
views
Textbook Question
In Exercises 1–22, solve the differential equation.
y' = eʸ/xy
1
views
Textbook Question
In Exercises 1–22, solve the differential equation.
y' = xy ln x ln y
1
views
Textbook Question
In Exercises 1–22, solve the differential equation.
x dy - (x⁴ - y) dx = 0
Textbook Question
In Exercises 1–22, solve the differential equation.
x dy + (3y - x⁻² cos x) dx = 0, x > 0