Question

Determine whether the following Differential equation is seperable, Linear, Exact, homogeneous, and Bernoulli. Next find its solution by respective method.

8 cos^2 ydx + csc^2 xdy=0

Answer 1

Rewrite in the form of a first order separable ODE

$$\dfrac{1}{\sin^2x}dy+8\cos^2ydx=0$$

$$\dfrac{1}{\cos^2y}dy=-8\sin^2xdx$$

Integrate

$$\int\dfrac{1}{\cos^2y}dy=-\int8\sin^2xdx$$

$$\tan y=-4\int(1-\cos(2x)dx$$

$$\tan y=-4x+2\sin(2x)+C$$

$$y=\tan^{-1}(-4x+2\sin(2x)+C)+\pi n, n\in\Z$$