Find a one parameter family of solutions of the following first order ordinary differential equation

polivijuye 2022-06-19 Answered
Find a one parameter family of solutions of the following first order ordinary differential equation
( 3 x 2 + 9 x y + 5 y 2 ) d x ( 6 x 2 + 4 x y ) d y = 0
Hello. So I am stuck after I find out that they are not exact. Please help.
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Answers (2)

plodno8n
Answered 2022-06-20 Author has 17 answers
The equation is
3 x 2 + 9 x y ( x ) + 5 y ( x ) 2 ( 6 x 2 + 4 x y ( x ) ) y ( x ) = 0
Looking at the last term, let y ( x ) = u ( x ) 3 2 x to get
4 x u ( x ) u ( x ) + 5 u ( x ) 2 + 3 x 2 4 = 0
that is to say
2 x ( u 2 ( x ) ) + 5 u 2 ( x ) + 3 x 2 4 = 0
So, let u ( x ) = ± v ( x ) to get
2 x v ( x ) + 5 v ( x ) + 3 x 2 4 = 0
which looks to be simple.

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telegrafyx
Answered 2022-06-21 Author has 8 answers
This is a homogeneous DE so making y = u x we obtain
x u = u ( u + 3 ) + 3 4 u + 6
which is separable giving
( 4 u + 6 ) d u u ( u + 3 ) + 3 = d x x
or
ln ( u ( u + 3 ) + 3 ) 2 = C 0 + ln x ( u ( u + 3 ) + 3 ) 2 = C 1 x
and finally
y = x 2 ( 3 ± C 2 x 3 )

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