Solve differential equation 2xy-9x^2+(2y+x^2+1)dy/dx=0, y(0)= -3

Chesley 2020-12-29 Answered
Solve differential equation 2xy9x2+(2y+x2+1)dy/dx=0, y(0)= -3
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2abehn
Answered 2020-12-30 Author has 88 answers

(2xy9x2)dx+(2y+x2+1)dy=0
Comparing with M dx + N dy = 0 we get
M=2xy9x2, N=2y+x2+1
(dM)/dy=2x0=2x
(dN)/dx=0+2x+0=2x
(dM)/dy=(dN)/dx
int Mdx+int (terms of N not containing x) dy=C
=>(2xy9x2)dx+(2y+1)dy=C
=>(2yx2/2)9x3/3+2y2/2+y=C
x2y3x3+y2+y=C
y2+(x2+1)y3x3=C (1)
Given y(0) = −3
substitute x=0 and y= -3 in (1) we get
(3)2+(02+1)(3)3(03)=C
9−3=C int C=6
substitute C=6 in (1) we get
y2+(x2+1)y3x3=6
y2+(x2+1)y3x36=0
2xy9x2+(2y+x2+1)dy/dx=0
y(0)=3isy2+(x2+1)y
3x36=0

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