I was messing around with a program I wrote to trace through slope fields using Euler's method, and I noticed this equation looked particularly funny: d y d x = y − x y ⋅ sin ⁡ ( y ) Is there any solution, implicit or explicit?

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I was messing around with a program I wrote to trace through slope fields using Euler&#039;s method, and I noticed this equation looked particularly funny:                    d        y                    d        x              =                    y        −        x                    y        ⋅        sin        ⁡        (        y        )            Is there any solution, implicit or explicit?

wijnvatid · Accepted Answer

d        y                    d        x              =                    y        −        x                    y        sin        ⁡        y              (  y  −  x  )                    d        x                    d        y              =  y  sin  ⁡  yLet   u  =  y  −  x,Then   x  =  y  −  u                    d        x                    d        y              =  1  −                    d        u                    d        y              ∴  u      (    1    −                            d          u                          d          y                      )    =  y  sin  ⁡  y  u  −  u                    d        u                    d        y              =  y  sin  ⁡  y  u                    d        u                    d        y              −  u  =  −  y  sin  ⁡  yThis belongs to an Abel equation of the second kind in the canonical form.

I was messing around with a program I wrote to trace through slope fields using Euler's method, and I noticed this equation looked particularly funny: dy/dx=y−x/y*sin(y)

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