Consider the differential equation $\frac { d y } { d x } = 2 x - y$. (a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated. (b) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ 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 $y = m x + b$ is a solution to the differential equation.
Consider the differential equation $\frac { d y } { d x } = 2 x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated.
(b) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ 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 $y = m x + b$ is a solution to the differential equation.