solve the following Exact differential equation

Question

Please solve the following Exact differential equation

( y³sin2x) dx - ( 3y² cos²x) dy = 0

please write the solution with steps

Answer 1

If you separate the variables, you have { sin(2x)/cos²x }dx = (3/y)dy

Using trig substitutions, we have (2/3)tanx dx = dy/y

This should be no problem from here, as -(2/3)ln(cosx) = ln(y) + ln(constant) = ln(cy)

Then, just simplify the expression (cosx) = (cy)^(-3/2)

^{(Note: Absolue value symbols left off of ln functions!)}