Solve the given differential equation by separation of variables. csc(y) dx + sec²(x) dy = 0

Solution:

Given, the differential equation is cosec (y) dx + sec²(x) dy = 0

We have to solve the equation by separation of variables.

The equation can be rewritten as

cosec (y) dx = -sec²(x) dy

We know, cosec y = 1/sin y and sec x = 1/cos x

\(\frac{1}{sin y}dx=\frac{-1}{cos^{2}x}dy\)

On cross multiplication,

cos²(x) dx = -siny dy

Taking integral,

\(\int cos^{2}x\: dx\, =\, \int -siny\, dy\)

We know, \(cos^{2}x=\frac{1+cos(2x)}{2}\)

Now, \(\int \frac{1+cos(2x)}{2} dx = \int -siny dy\)

On rearranging the terms,

\(\int \frac{1}{2}dx\, +\, \int \frac{1}{2}cos(2x)dx\, =\, \int -siny\, dy\)

\(\frac{1}{2}\int dx\, +\, \frac{1}{2}\int cos(2x)dx\, =\, \int -siny\, dy\)

\(\frac{1}{2}x\, +\, \frac{1}{2}\, \frac{sin(2x)}{2}\, =\, cosy\, +\, C\)

\(\frac{1}{2}x\, +\, \frac{sin(2x)}{4}\, =\, cosy\, +\, C\)

Therefore, the solution is \(\frac{1}{2}x\, +\, \frac{sin(2x)}{4}\, =\, cosy\, +\, C\)