How to solve this differential equation: x d y </mrow>

Quintacj 2022-05-24 Answered
How to solve this differential equation:
x d y d x = y + x e x e y ?
I tried to rearrange the equation to the form f ( y x ) but I couldn't thus I couldn't use v = y x to solve it.
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Answers (1)

Tristan Ward
Answered 2022-05-25 Author has 8 answers
x d y d x = y + x e x e y
x d y d x = y + x e x y
Let u = y x,
Then y = u + x
d y d x = d u d x + 1
x ( d u d x + 1 ) = u + x + x e u
x d u d x + x = u + x + x e u
x d u d x = x e u + u
( x e u + u ) d x d u = x
Let v = x + u e u ,
Then x = v u e u
d x d u = d v d u ( u + 1 ) e u
e u v ( d v d u ( u + 1 ) e u ) = v u e u
e u v d v d u ( u + 1 ) v = v u e u
e u v d v d u = ( u + 2 ) v u e u
v d v d u = ( u + 2 ) e u v u e 2 u
This belongs to an Abel equation of the second kind.
Let t = ( u + 1 ) e u ,
Then u = W ( e t ) 1
d v d u = d v d t d t d u = ( u + 2 ) e u d v d t
( u + 2 ) e u v d v d t = ( u + 2 ) e u v u e 2 u
v d v d t = v u e u u + 2
v d v d t v = t ( W ( e t ) 1 ) W ( e t ) ( W ( e t ) + 1 )
This belongs to an Abel equation of the second kind in the canonical form.
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