solve (cosx tany +cos(x+y))dx + (sin x sec 2 y+cos(x+y))dy=0

Question

differential equation  topic

solve (cosx tany +cos(x+y))dx + (sin x sec 2 y+cos(x+y))dy=0

Answer 1

This can be expanded:

cosxtany+cosxcosy-sinxsiny+

sinxsec²xdx/dx+cosxcosydy/dx-sinxsinydy/dx=0

Now rearrange the terms:

(cosxtany+sinxsec²ydy/dx)+

(-sinxsiny+cosxcosydy/dx)+

(cosxcosy-sinxsinydy/dx)=0.

Each of these pairs is a derivative:

d(sinxtany)/dx+d(cosxsiny)/dx+d(sinxcosy)/dx=0.

Now we can integrate:

sinxtany+cosxsiny+sinxcosy=C, constant of integration.

This can also be written: sinxtany+sin(x+y)=C.