What is the standard method for finding solutions of differential equations such as this one? (if th

Ellen Chang 2022-07-13 Answered
What is the standard method for finding solutions of differential equations such as this one? (if there is any)
x y = y 2 ( 2 x + 1 ) y + x 2 + 2 x
where y = a x + b is a particular solution.
Do I substitute y with a x + b + u ( x ) and then search for a solution or am I not noticing something and there's quicker way?
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Answers (1)

bap1287dg
Answered 2022-07-14 Author has 13 answers
I think that your hint is "there exists a , b such thaht y ( x ) = a x + b is solution". We find easily that y ( x ) = x and y ( x ) = x + 1 are solutions.
Hint: Note that your equation is
x y ( x ) x = ( y ( x ) x ) ( y ( x ) x 1 )
Now put y ( x ) = x + z ( x ).
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