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(a) Show that every member of the family of functions y = (6 In(x) + C)/x , x > 0, is a solution of the differential equation x²y' + xy = 6. (Simplify as much as possible.) 6 In(x) + C 6x · (1/x) – (6 In(x) +C) y' = y = x2 LHS = x?y' + xy = x2 . 6 In(x) + C + x: x2 + 6 In(x) + C = = RHS, so y is a solution of the differential equation.

(a) Show that every member of the family of functions y = (6 In(x) + C)/x , x > 0, is a solution of the differential equation x²y' + xy = 6. (Simplify as much as possible.) 6 In(x) + C 6x · (1/x) – (6 In(x) +C) y' = y = x2 LHS = x?y' + xy = x2 . 6 In(x) + C + x: x2 + 6 In(x) + C = = RHS, so y is a solution of the differential equation.

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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(a) Show that every member of the family of functions y = (6 In(x) + C)/x , x > 0, is a solution of the differential equation x?y' + xy = 6. (Simplify as much as possible.) %D 6 In(x) + C 6x · (1/x) – (6 In(x) +C) y' y = => x2 x2y' + xy = x2. 6 In(x) + C LHS = + x • x2 + 6 In(x) + C = RHS, %D so y is a solution of the differential equation.
Transcribed Image Text:(a) Show that every member of the family of functions y = (6 In(x) + C)/x , x > 0, is a solution of the differential equation x?y' + xy = 6. (Simplify as much as possible.) %D 6 In(x) + C 6x · (1/x) – (6 In(x) +C) y' y = => x2 x2y' + xy = x2. 6 In(x) + C LHS = + x • x2 + 6 In(x) + C = RHS, %D so y is a solution of the differential equation.
(c) Find a solution of the differential equation that satisfies the initial condition y(1) = 2. (Simplify as much as possible.) y = (d) Find a solution of the differential equation that satisfies the initial condition y(2) = 1. (Simplify as much as possible.) y =
Transcribed Image Text:(c) Find a solution of the differential equation that satisfies the initial condition y(1) = 2. (Simplify as much as possible.) y = (d) Find a solution of the differential equation that satisfies the initial condition y(2) = 1. (Simplify as much as possible.) y =
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