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 twelve points indicated, and sketch the solution curve that passes through the point $( 0, 1 )$. (b) The solution curve that passes through the point $( 0, 1 )$ has a local minimum at $x = \ln \left( \frac { 3 } { 2 } \right)$. What is the $y$-coordinate of this local minimum? (c) Let $y = f ( x )$ be the particular solution to the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f ( - 0.4 )$. Show the work that leads to your answer. (d) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Determine whether the approximation found in part (c) is less than or greater than $f ( - 0.4 )$. Explain your reasoning.
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 twelve points indicated, and sketch the solution curve that passes through the point $( 0, 1 )$.\\
(b) The solution curve that passes through the point $( 0, 1 )$ has a local minimum at $x = \ln \left( \frac { 3 } { 2 } \right)$. What is the $y$-coordinate of this local minimum?\\
(c) Let $y = f ( x )$ be the particular solution to the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f ( - 0.4 )$. Show the work that leads to your answer.\\
(d) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Determine whether the approximation found in part (c) is less than or greater than $f ( - 0.4 )$. Explain your reasoning.