## lgbasallote 3 years ago Find the general solution: $(3+y+2y^2\sin^2 x)dx + (x+ 2xy -y\sin 2x)dy=0$

1. lgbasallote

$\frac{\partial M}{\partial y} = 1 + 4y\sin^2 x$ $\frac{\partial N}{\partial x} = 1 + 2y - 2y\cos 2x$ so i guess tis isnt exact? or did i do something wrong?

2. A.Avinash_Goutham

wait wat's 1-cos2x?

3. A.Avinash_Goutham

it's 2sin^2x......it's exact

4. lgbasallote

wait what?

5. A.Avinash_Goutham

u kno 1-cos2x = 2sinx*sinx

6. lgbasallote

yeah?

7. lgbasallote

8. A.Avinash_Goutham

actually um wateva u call it it's root((1-cos2x)/2) = sinx

9. lgbasallote

so it's exact..

10. lgbasallote

$\int (3+y+2y^2 \sin^2 x)dx$ $3x + xy + y^2(x - \sin x \cos x)$

11. lgbasallote

then i take derivative of y

12. lgbasallote

$x + 2yx - 2y\sin x\cos x + g'(y) = x+2xy - y\sin 2x$ $g'(y) = 2y\sin x \cos x - y\sin 2x$ $g(y) = 2\sin x \cos x - \sin 2x$ so the G.S. would be $3x + xy + y^2(x - \sin x \cos x) + 2\sin x \cos x - \sin 2x + C$ correct?

13. A.Avinash_Goutham

u kno there's a direct solution for exact equations

14. lgbasallote

really? what?

15. A.Avinash_Goutham

i dont remember xactly but it's smthin llike integrate m(x,y) keeping y/x as a constant = integrate n(x,y) not containing terms of x/y + c

16. lgbasallote

oh that..yeah...im more comfortable with this though

17. lgbasallote

im more comfortable when the solution is one thread instead of inconsistent

18. A.Avinash_Goutham

ok