Question

Find the equation of the curve passing through the point $(0,\frac \pi4)$whose differential equation is, $sin\ x cos \ y\ dx+cos\ xsin\ y\ dy=0$

Answer 1

The differential equation of the given curve is:

$sin\ x cos \ y\ dx+cos\ xsin\ y\ dy=0$

⇒$\frac {sin\ x cos \ y\ dx+cos\ xsin\ y\ dy}{cos\ x cps\ y}=0$

⇒$tan\ x\ dx+tan \ y\ dy=0$

Integrating both sides, we get:

$log\ (sec\ x)+log\ (sec\ y)=log\ C$

$log\ (sec\ x.sec\ y)=log\ C$

⇒$sec\ x.sec\ y=C$ ...(1)

The curve passes through point $(0,\frac \pi4)$.

$∴1×\sqrt2=C$

⇒$C=\sqrt 2$

On substituting $C=\sqrt 2$ in equation (1), we get:

$sec\ x.sec \ y=\sqrt 2$

⇒$secx.\frac {1}{cos\ y}=\sqrt 2$

⇒$cos\ y=\frac {sec\ x}{\sqrt 2}$

Hence, the required equation of the curve is $cos\ y=\frac {sec\ x}{\sqrt 2}$.