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Classify each of the equations above as autonomous, separable, linear, homogeneous, exact, Bernoulli, function of a linear combination, or neither. A. y' = sin(x) + cos(y); B. y = +2 C. y = y; D. y' = ² + y² +eª²-y²; E. y' = x³y + y³x; F. y'=sin(x+2y-2) + cos(x +2y + 1); G. y' = 2y H. y' = x³ +y³r; I. (x² + 2xy)dx + (8x² + 5y²)dy = 0; J. y' = x³.

Classify each of the equations above as autonomous, separable, linear, homogeneous, exact, Bernoulli, function of a linear combination, or neither. A. y' = sin(x) + cos(y); B. y = +2 C. y = y; D. y' = ² + y² +eª²-y²; E. y' = x³y + y³x; F. y'=sin(x+2y-2) + cos(x +2y + 1); G. y' = 2y H. y' = x³ +y³r; I. (x² + 2xy)dx + (8x² + 5y²)dy = 0; J. y' = x³.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Classification of Differential Equations **Objective:** Classify each of the equations below as autonomous, separable, linear, homogeneous, exact, Bernoulli, function of a linear combination, or neither. Given Differential Equations: **A.** \( y' = \sin(x) + \cos(y); \) **B.** \( y' = \frac{y}{x} + \frac{2x}{y+2x}; \) **C.** \( y' = \frac{x^2 + y}{\ln(x)}; \) **D.** \( y' = e^{x^2 + y^2} + e^{x^2 - y^2}; \) **E.** \( y' = x^3 y + y^3 x; \) **F.** \( y' = \sin(x + 2y - 2) + \cos(x + 2y + 1); \) **G.** \( y' = \frac{y^2 + x^2}{2y}; \) **H.** \( y' = x^3 + y^3 x; \) **I.** \( (x^2 + 2xy)dx + (8x^2 + 5y^2)dy = 0; \) **J.** \( y' = x^3; \) ### Explanation of Terms: - **Autonomous:** Differential equations where the derivative \( y' \) depends only on the variable \( y \). - **Separable:** Differential equations that can be written as the product of a function of \( x \) and a function of \( y \). - **Linear:** Differential equations that can be written in the form \( y' + p(x)y = q(x) \). - **Homogeneous:** Differential equations where every term is a function of \( x \) and \( y \) with the same degree. - **Exact:** Differential equations that can be derived from a potential function \( \Phi(x, y) \) such that \( \frac{\partial \Phi}{\partial x} = M \) and \( \frac{\partial \Phi}{\partial y} = N \). - **Bernoulli:** Differential equations of the form \( y' + p(x)y = q(x)y^n \). - **Function of a Linear Combination:** \(
Transcribed Image Text:### Classification of Differential Equations **Objective:** Classify each of the equations below as autonomous, separable, linear, homogeneous, exact, Bernoulli, function of a linear combination, or neither. Given Differential Equations: **A.** \( y' = \sin(x) + \cos(y); \) **B.** \( y' = \frac{y}{x} + \frac{2x}{y+2x}; \) **C.** \( y' = \frac{x^2 + y}{\ln(x)}; \) **D.** \( y' = e^{x^2 + y^2} + e^{x^2 - y^2}; \) **E.** \( y' = x^3 y + y^3 x; \) **F.** \( y' = \sin(x + 2y - 2) + \cos(x + 2y + 1); \) **G.** \( y' = \frac{y^2 + x^2}{2y}; \) **H.** \( y' = x^3 + y^3 x; \) **I.** \( (x^2 + 2xy)dx + (8x^2 + 5y^2)dy = 0; \) **J.** \( y' = x^3; \) ### Explanation of Terms: - **Autonomous:** Differential equations where the derivative \( y' \) depends only on the variable \( y \). - **Separable:** Differential equations that can be written as the product of a function of \( x \) and a function of \( y \). - **Linear:** Differential equations that can be written in the form \( y' + p(x)y = q(x) \). - **Homogeneous:** Differential equations where every term is a function of \( x \) and \( y \) with the same degree. - **Exact:** Differential equations that can be derived from a potential function \( \Phi(x, y) \) such that \( \frac{\partial \Phi}{\partial x} = M \) and \( \frac{\partial \Phi}{\partial y} = N \). - **Bernoulli:** Differential equations of the form \( y' + p(x)y = q(x)y^n \). - **Function of a Linear Combination:** \(
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