Question

[(xy) Cos(xy) + sin(xy)] dx + x²Cos(xy) dy
=0​

Answer 1

Answer:

x sin(xy) = C

Step-by-step explanation:

It looks like a differential of sin(xy) might play a role, so start there to see

d( sin(xy) ) = cos(xy) d(xy)

  = cos(xy) ( y dx + x dy )

  = y cos(xy) dx + x cos(xy) dy

That looks pretty good, but we need that multiplied by x, and at the same time, we need the extra sin(xy) dx term, so we tweak this a bit to get

d( x sin(xy) ) = sin(xy) dx + x d( sin(xy) )

  = sin(xy) dx + xy cos(xy) dx + x² cos(xy) dy

and that's exactly the expression in the given equation.  So the equation is

d( x sin(xy) ) = 0

from which we conclude that

x sin(xy) = C

for some constant C.

Hope that helps.