Find the general solution of the following DE. xy(y')² + (x² + y²)y' + xy = 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 5T
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Transcribed Image Text:Find the general solution of the following DE.
xy(y')² + (x² + y²)y' + xy = 0
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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