I am trying to solve the differential equation y ′ ( x ) − y ( x ) 2 x = x sin ⁡ ( x y ( x ) ) I think it is separable variable differential equations. I tried to substitute: z = x y and y ′ = z − z ′ x z 2 z − z ′ x z 2 − 1 2 z = x sin ⁡ ( z )multiply by z 2 z − z ′ x − z 2 = z 2 x sin ⁡ ( z )And now I have no idea how to manipulate this.

Name: I am trying to solve the differential equation y ′ ( x ) − y ( x ) 2 x = x sin ⁡ ( x y ( x ) ) I think it is separable variable differential equations. I tried to substitute: z = x y and y ′ = z − z ′ x z 2 z − z ′ x z 2 − 1 2 z = x sin ⁡ ( z )multiply by z 2 z − z ′ x − z 2 = z 2 x sin ⁡ ( z )And now I have no idea how to manipulate this.
Uploaded: 2022-02-23T17:22:57+00:00
Description: I am trying to solve the differential equation y ′ ( x ) − y ( x ) 2 x = x sin ⁡ ( x y ( x ) ) I think it is separable variable differential equations. I tried to substitute: z = x y and y ′ = z − z ′ x z 2 z − z ′ x z 2 − 1 2 z = x sin ⁡ ( z )multiply by z 2 z − z ′ x − z 2 = z 2 x sin ⁡ ( z )And now I have no idea how to manipulate this.

Question

I am trying to solve the differential equation      y    ′        (    x    )    −            y              (        x        )                    2      x        =  x  sin  ⁡      (          x              y                  (          x          )                      )  I think it is separable variable differential equations. I tried to substitute:  z  =      x    y  and      y    ′    =            z      −              z        ′            x              z      2                  z      −              z        ′            x              z      2        −      1          2      z        =  x  sin  ⁡  (  z  )multiply by       z    2    z  −      z    ′    x  −      z    2    =      z    2    x  sin  ⁡  (  z  )And now I have no idea how to manipulate this.

Hadley Cunningham · Accepted Answer

HintLet   t  =            y      x      ,Then   y  =  x  t                    d        y                    d        x              =  t  +  x                    d        t                    d        x              ∴  t  +  x                    d        t                    d        x              −            t      2        =  x  sin  ⁡            1      t        x                    d        t                    d        x              =  x  sin  ⁡            1      t        −            t      2            (    x    sin    ⁡                  1        t              −                  t        2              )                      d        x                    d        t              =  xThis belongs to an Abel equation of the second kind.

I am trying to solve the differential equation y ′ ( x ) − y...

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