Solution of differential equation (2 x cos y+y2 cos x) d x+(2 y sin x-x2 sin y) d y=0 is

Question

Solution of differential equation (2 x cos y+y2 cos x) d x+(2 y sin x-x2 sin y) d y=0 is (A) x2 cos y+y2 sin x=C (B) x cos y-y sin x=C (C) x2 cos 2 y+y2 sin 2 x=C (D) x cos y+y sin x=C

Tardigrade Admin · Accepted Answer

2 x cos y d x+y2 cos x d x+2 y sin x d y-x2 sin y d y=0 ⇒ (2 x cos y d x-x2 sin y d y)+ (y2 cos x d x+2 y sin x d y)=0 ⇒ d(x2 cos y+y2 sin x)=0 ⇒ x2 cos y+y2 sin x=C

Q. Solution of differential equation $(2 x cos y + y^{2} cos x) d x + (2 y sin x - x^{2} sin y) d y = 0$ is

$x^{2} cos y + y^{2} sin x = C$

$x cos y - y sin x = C$

$x^{2} cos^{2} y + y^{2} sin^{2} x = C$

$x cos y + y sin x = C$

$2 x cos y d x + y^{2} cos x d x + 2 y sin x d y - x^{2} sin y d y = 0$
$\Rightarrow (2 x cos y d x - x^{2} sin y d y) +$
$(y^{2} cos x d x + 2 y sin x d y) = 0$
$\Rightarrow d (x^{2} cos y + y^{2} sin x) = 0$
$\Rightarrow x^{2} cos y + y^{2} sin x = C$

Q. Solution of differential equation (2xcosy+y2cosx)dx+(2ysinx−x2siny)dy=0 is

x2cosy+y2sinx=C

xcosy−ysinx=C

x2cos2y+y2sin2x=C

xcosy+ysinx=C