Solve this differential equation ( 2 x + y ) d x + ( x − 2 y ) d y = 0as an exact differential equation and I know it's exact because I solve the equaliy ∂ ( 2 x + y ) ∂ y = 1and ∂ ( x − 2 y ) ∂ x = 1so following the steps to solve this kind of equations i have: x 2 + g ′ ( y ) = x − 2 yand 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

Question

Solve this differential equation  (  2  x  +  y  )  d  x  +  (  x  −  2  y  )  d  y  =  0as an exact differential equation and I know it&#039;s exact because I solve the equaliy            ∂      (      2      x      +      y      )              ∂      y        =  1and            ∂      (      x      −      2      y      )              ∂      x        =  1so following the steps to solve this kind of equations i have:      x    2    +      g    ′    (  y  )  =  x  −  2  yand      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&#039;ll apreciate

Alexandria Rubio · Accepted Answer

Step 1The solution is   F  (  x  ,  y  )  =  C such that            ∂      F              ∂      x        =  2  x  +  y            ∂      F              ∂      y        =  x  −  2  yTreating 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 2Treating x as constant, take the partial w.r.t. y of the above expression            ∂      F              ∂      y        =  x  +      g    ′    (  y  )  =  x  −  2  yThis 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

solve this differential equation (2x+y)dx+(x−2y)dy=0

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