Question

Find the general solution of the differential equation: cosx (1 + cosy) dx - siny (1 + sinx) dy = 0

Answer 1

To solve the differential equation, we can start by rearranging it:

cos(x) (1 + cos(y)) dx = sin(y) (1 + sin(x)) dy

Now, let's divide both sides of the equation by cos(x) (1 + cos(y)) and rearrange the terms:

dx/dy = (sin(y) (1 + sin(x))) / (cos(x) (1 + cos(y)))

To simplify the equation further, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1:

dx/dy = (sin(y) (1 + sin(x))) / (cos(x) (1 + cos(y))) * (sin^2(x) + cos^2(x)) / (sin^2(x) + cos^2(x))

dx/dy = sin(y) (1 + sin(x)) sin^2(x) / (cos(x) (1 + cos(y)) (sin^2(x) + cos^2(x)))

Now, let's simplify the expression even more:

dx/dy = sin(y) sin(x) (1 + sin(x)) / (cos(x) (1 + cos(y)))

dx/dy = (sin(y) sin(x) (1 + sin(x))) / (cos(x) (1 + cos(y)))

Now, let's separate the variables by multiplying both sides by dy:

dy = (cos(x) (1 + cos(y))) / (sin(y) sin(x) (1 + sin(x))) dx

dy = (cos(x) / (sin(x))) (1 + cos(y)) / (sin(y) (1 + sin(x))) dx

Now, we can integrate both sides:

∫ dy = ∫ (cos(x) / sin(x)) (1 + cos(y)) / (sin(y) (1 + sin(x))) dx

y = ∫ (cos(x) / sin(x)) (1 + cos(y)) / (sin(y) (1 + sin(x))) dx + C

Unfortunately, the integral on the right side does not have a simple closed-form solution. Therefore, the general solution of the given differential equation is:

y = ∫ (cos(x) / sin(x)) (1 + cos(y)) / (sin(y) (1 + sin(x))) dx + C

where C is the constant of integration.