Consider the differential equation $\frac { d y } { d x } = x ^ { 2 } - \frac { 1 } { 2 } y$. (a) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. (b) Let $y = f ( x )$ be the particular solution to the given differential equation whose graph passes through the point $( - 2,8 )$. Does the graph of $f$ have a relative minimum, a relative maximum, or neither at the point $( - 2,8 )$ ? Justify your answer. (c) Let $y = g ( x )$ be the particular solution to the given differential equation with $g ( - 1 ) = 2$. Find $\lim _ { x \rightarrow - 1 } \left( \frac { g ( x ) - 2 } { 3 ( x + 1 ) ^ { 2 } } \right)$. Show the work that leads to your answer. (d) Let $y = h ( x )$ be the particular solution to the given differential equation with $h ( 0 ) = 2$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $h ( 1 )$.
Consider the differential equation $\frac { d y } { d x } = x ^ { 2 } - \frac { 1 } { 2 } y$.\\
(a) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$.\\
(b) Let $y = f ( x )$ be the particular solution to the given differential equation whose graph passes through the point $( - 2,8 )$. Does the graph of $f$ have a relative minimum, a relative maximum, or neither at the point $( - 2,8 )$ ? Justify your answer.\\
(c) Let $y = g ( x )$ be the particular solution to the given differential equation with $g ( - 1 ) = 2$. Find $\lim _ { x \rightarrow - 1 } \left( \frac { g ( x ) - 2 } { 3 ( x + 1 ) ^ { 2 } } \right)$. Show the work that leads to your answer.\\
(d) Let $y = h ( x )$ be the particular solution to the given differential equation with $h ( 0 ) = 2$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $h ( 1 )$.