How to prove an ordinary differential equation is not separable?Some elementary exercises require one to determine whether or not an ordinary differential equation is separable. For example, it is understood that the equation y ′ = 1 − 2 x y x 2 is not separable. An easier example is y ′ = x + y. Usually this is intuitive understandable; however, can one give a strict proof that the right hand of these equations cannot be written as a product of the form p ( y ) q ( x )?

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How to prove an ordinary differential equation is not separable?Some elementary exercises require one to determine whether or not an ordinary differential equation is separable. For example, it is understood that the equation      y    ′    =            1      −      2      x      y              x      2      is not separable. An easier example is       y    ′    =  x  +  y. Usually this is intuitive understandable; however, can one give a strict proof that the right hand of these equations cannot be written as a product of the form   p  (  y  )  q  (  x  )?

lefeuilleton42 · Accepted Answer

In general, a             C        2   function   f  (  x  ,  y  )  ≠  0 can be expressed in the form   p  (  x  )  q  (  y  ) if and only if                    ∂        2            ln      ⁡      f              ∂      x      ∂      y        =  0.Proof: First, note that if   f  (  x  ,  y  )  =  p  (  x  )  q  (  y  ), we have                    ∂        2            ln      ⁡      f              ∂      x      ∂      y        =      ∂          ∂      x            (          1      f                      ∂        f                    ∂        y              )    =      ∂          ∂      x            (                  p        (        x        )                  q          ′                (        y        )                    p        (        x        )        q        (        y        )              )    =      ∂          ∂      x            (                            q          ′                (        y        )                    q        (        y        )              )    =  0.To prove the converse, suppose that                    ∂        2            ln      ⁡      f              ∂      x      ∂      y        =      ∂          ∂      x            (                  ∂        ln        ⁡        f                    ∂        y              )    =  0.If this is true, the quantity in square brackets must be a function of y only:            ∂      ln      ⁡      f              ∂      y        =  g  (  y  )  .We can then integrate this with respect to   y; if   G  (  y  ) is the antiderivative of   g  (  y  ), we must have  ln  ⁡  f  =  G  (  y  )  +  h  (  x  )  .(Note that we are free to add a &quot;constant with respect to y&quot;, i.e., a function h that depends only on x, to the antiderivative.) We conclude that  f  (  x  ,  y  )  =      e          G      (      y      )      +      h      (      x      )        =  p  (  x  )  q  (  y  )where   p  (  x  )  =      e          h      (      x      )       and   q  (  y  )  =      e          G      (      y      )      (I have implicitly assumed throughout that   f  (  x  ,  y  )  &amp;gt;  0, but the above proof can be easily extended to domains where   f  (  x  ,  y  )  &amp;lt;  0 as well.)

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