(2xy - 3x^{2})dx + (x^{2} + y)dy = 0

Donald Johnson 2021-12-21 Answered
(2xy3x2)dx+(x2+y)dy=0
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Expert Answer

Stella Calderon
Answered 2021-12-22 Author has 35 answers
The given differential equation is
(2xy3x2)dx+(x2+y)dy=0
Comparing it with Mdx+Mdy=0
M=2xy3x2 N=x2+y
dMdy=2x dNdx=2x
Here, dMdy=dNdx
Then, it is exact differential equation, so solution is
Mdx+N (term not contining x) dy=c
(2x3x2)dx+ydy=c
y2x223x33+y22=c
x2yx3+y22=c
This is solution of given DE.
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Charles Benedict
Answered 2021-12-23 Author has 32 answers
Assuming (2xy3x2)dx+(x2+y)dy=0
Then 2xy+(x2+y)dydx=3x2 (1)
Notice that dx2y+y22dx is:
x2dydx+2xy+2y2dydx
=2xy+(x2+y)dydx which is left hand side of (1)
So integrating (1) gives
x2y+y22=x3+k
So y2+2x2y2x3+c=0 is the solution
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RizerMix
Answered 2021-12-29 Author has 438 answers

Let M=(2xy3x2) so that DM/Dy=2x
Let N=:(x2+y) so that DN/Dx=2x
since DM/Dy=DN/Dx here D is to be read as delta.
The given Equation
(2xy3x2)dx+(x2+y)dy=0 is exact
Integrating yx2x6x+1/2y2=0
or, 2yx2x12x+y2=0

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