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 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 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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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
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
Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?
Textbook Question
In Exercises 1–22, solve the differential equation.
xy' + 2y = 1 - x⁻¹
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
1
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Textbook Question
In Exercises 1–22, solve the differential equation.
x dy - (x⁴ - y) dx = 0