Solve dy/dx − 2y tan x = y² tan x

dy/dx − 2y tan x = y² tan x

This is Bernoulli's equation with n = 2.

Reduce to linear form

Divide by y²:

y⁻² dy/dx − 2y⁻¹ tan x = tan x

Put z = 1/y, so dz/dx = −y⁻² dy/dx:

−dz/dx − 2z tan x = tan x → dz/dx + 2 tan x · z = −tan x

Integrating factor

IF = e^(2∫tan x dx) = e^(2 log sec x) = sec²x

Solution

z sec²x = −∫ tan x sec²x dx = −tan²x/2 + c

z = c cos²x − (1/2) tan²x cos²x = c cos²x − (1/2) sin²x

Replace z = 1/y:

1/y = c cos²x − ½ sin²x

or 2 = y(C cos²x − sin²x) with C = 2c.