Question
The equation of the curve passing through the point $$\left( {0,\,\frac{\pi }{4}} \right)$$ whose differential equation is $$\sin \,x\,\cos \,y\,dx + \cos \,x\,\sin \,y\,dy = 0,$$ is :
A.
$$\sec \,x\,\sec \,y = \sqrt 2 $$
B.
$$\cos \,x\,\cos \,y = \sqrt 2 $$
C.
$$\sec \,x = \sqrt 2 \,\cos \,y$$
D.
$$\cos \,y = \sqrt 2 \,\sec \,y$$
Answer :
$$\sec \,x\,\sec \,y = \sqrt 2 $$
Solution :
The given differential equation is $$\sin \,x\,\cos \,y\,dx + \cos \,x\,\sin \,y\,dy = 0$$
dividing by $$\cos \,x\,\cos \,y$$
$$ \Rightarrow \frac{{\sin \,x}}{{\cos \,x}}dx + \frac{{\sin \,y}}{{\cos \,y}}dy = 0$$
Integrating,
$$\eqalign{
& \int {\tan \,x\,dx} + \int {\tan \,y\,dy} = \log \,c \cr
& {\text{or }}\log \,\sec \,x\,\sec \,y = \log \,c \cr
& {\text{or }}\sec \,x\,\sec \,y = c \cr} $$
curve passes through the point $$\left( {0,\,\frac{\pi }{4}} \right)$$
$$\sec \,0\,\sec \frac{\pi }{4} = c = \sqrt 2 $$
Hence, the required equation of the curve is $$\sec \,x\,\sec \,y = \sqrt 2 .$$