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Find the general solution of the following DE. xy(y')² + (x² + y²)y' + xy = 0

Find the general solution of the following DE. xy(y')² + (x² + y²)y' + xy = 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the general solution of the following DE. xy(y')² + (x² + y²)y' + xy = 0
Transcribed Image Text:Find the general solution of the following DE. xy(y')² + (x² + y²)y' + xy = 0
Example 2: Solve x(y)-(3x²-1)y - 3x = 0 Solution: Using the quadratic formula to factor the left-hand side of the given differential equation, we identify: A=x. B=-(3x²-1) and C=-3x, and that (3x-1)=√(3x-1)-4x(-3x) 3x³-11√9x¹-6x² +1+12x³ 3x² -1± √9x² +6x² +1 3x²-1±√3x² +1} x² −1± √(3x² +1) 2x 2x 3x-1(3x+1) 3x²-1+3x²+1 3x²-1-3x²-1 y', 6x² -2 1 - 3x 2x Thus, the given differential equation can be expressed as (y - 3x)(y + +3)=0 Equating each factor to zero and solving the resulting differential equations, we have: -0 y-3x-0 dy-3xdx-0 3x² y'- y. y 3x² dx y+hx-C₂ y--hx+c₂
Transcribed Image Text:Example 2: Solve x(y)-(3x²-1)y - 3x = 0 Solution: Using the quadratic formula to factor the left-hand side of the given differential equation, we identify: A=x. B=-(3x²-1) and C=-3x, and that (3x-1)=√(3x-1)-4x(-3x) 3x³-11√9x¹-6x² +1+12x³ 3x² -1± √9x² +6x² +1 3x²-1±√3x² +1} x² −1± √(3x² +1) 2x 2x 3x-1(3x+1) 3x²-1+3x²+1 3x²-1-3x²-1 y', 6x² -2 1 - 3x 2x Thus, the given differential equation can be expressed as (y - 3x)(y + +3)=0 Equating each factor to zero and solving the resulting differential equations, we have: -0 y-3x-0 dy-3xdx-0 3x² y'- y. y 3x² dx y+hx-C₂ y--hx+c₂
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