Textbook Question
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
y' = y/x + cos ((y-x)/x)
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Ch. 9 - First-Order Differential Equations
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 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.AAE.10
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
(x sin y/x - y cos y/x)dx + (x cos y/x) dy = 0
Verified step by step guidance1
Identify the given differential equation: \(\left(x \sin \frac{y}{x} - y \cos \frac{y}{x}\right) dx + \left(x \cos \frac{y}{x}\right) dy = 0\).
Rewrite the equation in the form \(M(x,y) dx + N(x,y) dy = 0\), where \(M(x,y) = x \sin \frac{y}{x} - y \cos \frac{y}{x}\) and \(N(x,y) = x \cos \frac{y}{x}\).
Check if the equation is homogeneous by verifying if \(M(tx, ty) = t^n M(x,y)\) and \(N(tx, ty) = t^n N(x,y)\) for some degree \(n\). Since the functions depend on \(\frac{y}{x}\), this suggests homogeneity of degree 1.
Use the substitution \(v = \frac{y}{x}\), which implies \(y = vx\) and \(dy = v dx + x dv\) to transform the equation into a separable form in terms of \(v\) and \(x\).
Substitute \(y\) and \(dy\) into the original equation, simplify, and then separate variables to prepare for integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Homogeneous Differential Equations
A differential equation is homogeneous if it can be expressed so that each term is a function of the ratio y/x. This allows substitution like v = y/x to simplify the equation into a separable form, making it easier to solve.
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Classifying Differential Equations
Substitution Method (v = y/x)
By substituting v = y/x, the original variables are transformed, reducing the equation to one involving v and x only. This substitution leverages the homogeneity property and helps convert the differential equation into a separable or integrable form.
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Separable Differential Equations
Once the substitution is made, the resulting equation often becomes separable, meaning it can be written as a product of a function of v and a function of x. This allows integration on both sides separately to find the solution.
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Related Practice
Textbook Question
Use Euler’s method with dx = 0.2 to estimate y(1) if y′ = y and y(0) = 1. What is the exact value of y(1)?
Textbook Question
In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.
y' = 2y²(x-1), y(2) = -1/2, dx = 0.1, x* = 3
Textbook Question
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
(x²+y²)dx + xy dy = 0
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Textbook Question
Show that (0, 0) and (c/d, a/b) are equilibrium points. Explain the meaning of each of these points.
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Textbook Question
Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.
(x.exp(y/x) + y)dx - x dy = 0
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