Problem

Consider the differential equation $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = y - 2x$.

(a) Suppose that $f(x) = mx + b$ is a solution to the differential equation. Find the values of $m$ and $b$.

(b) Find the most general solution using the formula for linear differential equations: $y' + P(x)y = Q(x) \implies y = \frac{1}{I(x)} \left( \int I(x) Q(x) \mathrm{d}x + C \right)$ where $I(x) = \exp \left( \int P(x) \mathrm{d}x \right)$.