Question

The solution of the differential equation $x^4\frac{dy}{dx}+x^{3}y+cosec\left(xy\right)=0$ is equal to

• A. $2\cos \left(xy\right)+x^{-2}=C$
• B. $2\cos \left(xy\right)+y^{-2}=C$
• C. $2\sin \left(xy\right)+x^{-2}=C$
• D. $2\sin \left(xy\right)+y^{-2}+C$

Answer 1

Answer: (A)

$2\cos \left(xy\right)+x^{-2}=C$

Given,

$x^4\frac{dy}{dx}+x^{3}y+cosec\left(xy\right)=0$

Put $xy=t$

$y+x\frac{dy}{dx}=\frac{dt}{dx}$

Substituting it we get;

$x^3(\frac{dt}{dx}-\frac{t}{x})+x^{2}t+cosec\left(t\right)=0$

$\Rightarrow x^3\frac{dt}{dx}+cosec\left(t\right)=0$

$\Rightarrow \sin \left(t\right)dt+x^{-3}dx=0$

After Integrating we get;

$2\cos \left(xy\right)+x^{-2}=C$