Question

How to you find the general solution of `dy/dx=tan^2x`?

Answer 1

`y = tanx - x + C`, (where `C` is an arbitrary constant).

Explanation:

We have:

`dy/dx = tan^2x`

This is a First Order separable Differential Equation, so we can just collect terms in `y`, and terms in `x` and "separate the variables" to get:

`int \ dy = int \ tan^2x \ dx`

We can now integrate, and deal with the RHS integral by using the trig identify `tan^2 theta = sec^2 theta - 1`, so we get:

`\ \ \ \ \ y = int \ (sec^2x - 1) \ dx`
`:. y = tanx - x + C`, (where `C` is an arbitrary constant).