# Example One: v = ax + by + c

**new**expression for y. $$\dfrac{dy}{dx} = \dfrac{dv}{dx} - 1$$ Now, we set the two expressions equal. $$(x + y + 3)^2 \to v^2 = \dfrac{dv}{dx} - 1$$ $$\dfrac{dv}{dx} = v^2 + 1$$ $$\int\dfrac{1}{v^2 + 1}\, dv = \int 1\, dx$$ $$\arctan{v} = x + C \rightarrow v = \tan(x + C)$$ $$x + y + 3 = \tan(x + C) \to y = \tan(x + C) - x - 3$$

# Example Two: v = y/x

# Example Three: v = y^(1 - n)

David Witten