Question

Solution of the differential equation $({3xy}^2+xsin\left(xy\right))dy+(y^3+ysin\left(xy\right))dx=0$ is:

• A. ${xy}^3-\cos xy=c$
• B. ${xy}^3+\cos xy=c$
• C. ${xy}^2-\cos xy=c$
• D. ${xy}^2+\sin xy=c$

Answer 1

The correct option is A ${xy}^3-\cos xy=c$

Given,

$(3x{y}^2+xsin(xy\left)\right)dy+(y^3+ysin(xy\left)\right)dx=0$

$3x{y}^{2}dy+y^{3}dx+xsin\left(xy\right)dy+ysin\left(xy\right)dx=0$

$\to d\left(x{y}^3\right)+d(-cos(xy\left)\right)=0$

Integrate Both the sides

$\int d\left(x{y}^3\right)+\int d(-cos(xy\left)\right)=0$

$\to x{y}^3-cos\left(xy\right)=c$