Aldo Ashley

Aldo Ashley

Answered

2022-10-25

Solve this differential equation
( 2 x + y ) d x + ( x 2 y ) d y = 0
as an exact differential equation and I know it's exact because I solve the equaliy
( 2 x + y ) y = 1
and
( x 2 y ) x = 1
so following the steps to solve this kind of equations i have:
x 2 + g ( y ) = x 2 y
and
g ( y ) = x 2 y x 2
to be honest I have many doubts what are the next steps so if you can guide me I'll apreciate

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Answer & Explanation

Alexandria Rubio

Alexandria Rubio

Expert

2022-10-26Added 10 answers

Step 1
The solution is F ( x , y ) = C such that
F x = 2 x + y
F y = x 2 y
Treating y as constant, integrate the first equation with respect to x
F ( x , y ) = ( 2 x + y ) d x = x 2 + x y + g ( y )
Step 2
Treating x as constant, take the partial w.r.t. y of the above expression
F y = x + g ( y ) = x 2 y
This gives
g ( y ) = 2 y g ( y ) = y 2
F ( x , y ) = x 2 + x y y 2
so the solution is
x 2 + x y y 2 = C

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