Problem with first-order nonlinear ordinary differential equation

bobey

i have problem to find the solution for : (3x3y+2xy+y3)+(x2+y2)dy/dx=0

i have tried the exact equation method :

(3x3y+2xy+y3)dx+(x2+y2)dy=0

thus M(x,y)=(3x3y+2xy+y3)

and N(x,y)= (x2+y2)

then deltaM/deltay=3x3+2x+3y2
and deltaN/deltax=2x

Since deltaM/deltay does not equal to deltaN/deltax, this imply that the eqution is not exact

thus, finding/searching for integrating factor :

1. 1/N(deltaM/deltay-deltaN/deltax)=(3x3+3y3)/(3x3y+2xy+y3)

y cannot be eliminated . thus, this is a function of both x and y, not just x

2. 1/M(deltaN/deltax-deltaM/deltay)=(3x3+3y2)/(x2+y2)

x cannot be eliminated . thus, this is a function of both x and y, not just y

thus i cannot find the integrating factor in order to solve the DE. where i'm gone wrong? can somebody point it out? i guess may be in algebra...

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kosovtsov

You are on right direction. But not all of ODEs have an easy integrating factor.

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