Consider the differential equation $\dfrac{dy}{dx} = 2x - y$. (a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated. (b) Find $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer. (c) Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(2) = 3$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 2$? Justify your answer. (d) Find the values of the constants $m$ and $b$ for which $y = mx + b$ is a solution to the differential equation.
Consider the differential equation $\dfrac{dy}{dx} = 2x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated.
(b) Find $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer.
(c) Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(2) = 3$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 2$? Justify your answer.
(d) Find the values of the constants $m$ and $b$ for which $y = mx + b$ is a solution to the differential equation.