Consider the differential equation $\frac { d y } { d x } = \frac { 1 } { 2 } x + y - 1$. (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the axes provided in the exam booklet.) (b) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Describe the region in the $x y$-plane in which all solution curves to the differential equation are concave up. (c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = 1$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 0$ ? 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.
076$ because $f ( t ) < g ( t )$ for $0 \leq t < 1.617$ and $3 < t < 5.076$.
Consider the differential equation $\frac { d y } { d x } = \frac { 1 } { 2 } x + y - 1$.
(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.
(Note: Use the axes provided in the exam booklet.)
(b) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Describe the region in the $x y$-plane in which all solution curves to the differential equation are concave up.
(c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = 1$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 0$ ? 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.