A curve passes through $(0, 3)$, and its slope at every point $(x, y)$ is $y - 2x$. What is $y$ on the curve when $x = 1$?

Explanation

1. 01

   $\frac{dy}{dx} - y = -2x$ has the factor $e^{-x}$: $ye^{-x} = \int -2xe^{-x}\,dx = 2(x + 1)e^{-x} + C$.

2. 02

   So $y = 2x + 2 + Ce^{x}$; at $(0, 3)$, $2 + C = 3$, so $C = 1$.

3. 03

   At $x = 1$: $y = 2 + 2 + e = 4 + e$.