4, xydx (x² + 3y²) dy = 0 Homogeneous, 2nd degree Let x = vy dx= vdy + ydv Substitute. vy² (vdy+ydv)-(v²y2 + 3y2) dy=0 vy² (vdy + ydv)- y² (v²+3) dy = 0. divide by y2, v²dy + vydv (v²+3) dy = 0 vydv - 3dy = 0 divide by y. vdv - but : 3 dy y fvdv - 3 fo ²-3 ln y = -3 Inc v²-6lny = -6 Inc y² = 6lny - 6 Inc v² = = 0 6 In v = ²=6 = 6 In y/c x² = 6y² In y/c ans.
4, xydx (x² + 3y²) dy = 0 Homogeneous, 2nd degree Let x = vy dx= vdy + ydv Substitute. vy² (vdy+ydv)-(v²y2 + 3y2) dy=0 vy² (vdy + ydv)- y² (v²+3) dy = 0. divide by y2, v²dy + vydv (v²+3) dy = 0 vydv - 3dy = 0 divide by y. vdv - but : 3 dy y fvdv - 3 fo ²-3 ln y = -3 Inc v²-6lny = -6 Inc y² = 6lny - 6 Inc v² = = 0 6 In v = ²=6 = 6 In y/c x² = 6y² In y/c ans.
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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Question
where does the -3ln C comes from?
![4, xydx (x² + 3y²) dy = 0
Homogeneous, 2nd degree
Let
x = vy
dx= vdy + ydv
Substitute,
vy² (vdy+ydv)-(v²y2 + 3y2) dy=O
vy² (vdy + ydv) - y² (v²+ 3) dy = 0.
divide by y2,
v²dy + vydv (v²+3) dy = 0
vy dv 3 dy = 0
-
divide by y.
vdv
-
3 dy
y
= 0
fvdv3fd-fo
²-3 ln y = -3 in C
v²-6lny = -6 Inc
y² =
6lny - 6 In c
v² =
6 In
but : V =
= = 6 in 9/c
x² = 6y² In y/c
ans.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf96b0f3-b838-48d1-ae93-75213d32bb53%2F8b2fa164-6b77-422d-a3d5-7aa07440df47%2Fo1nmvi8_processed.png&w=3840&q=75)
Transcribed Image Text:4, xydx (x² + 3y²) dy = 0
Homogeneous, 2nd degree
Let
x = vy
dx= vdy + ydv
Substitute,
vy² (vdy+ydv)-(v²y2 + 3y2) dy=O
vy² (vdy + ydv) - y² (v²+ 3) dy = 0.
divide by y2,
v²dy + vydv (v²+3) dy = 0
vy dv 3 dy = 0
-
divide by y.
vdv
-
3 dy
y
= 0
fvdv3fd-fo
²-3 ln y = -3 in C
v²-6lny = -6 Inc
y² =
6lny - 6 In c
v² =
6 In
but : V =
= = 6 in 9/c
x² = 6y² In y/c
ans.
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