Consider the differential equation $\frac { d y } { d x } = 2 y - 4 x$. (a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through the point $( 0, 1 )$ and sketch the solution curve that passes through the point $( 0 , - 1 )$. (b) Let $f$ be the function that satisfies the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with a step size of 0.1, to approximate $f ( 0.2 )$. Show the work that leads to your answer. (c) Find the value of $b$ for which $y = 2 x + b$ is a solution to the given differential equation. Justify your answer. (d) Let $g$ be the function that satisfies the given differential equation with the initial condition $g ( 0 ) = 0$. Does the graph of $g$ have a local extremum at the point $( 0, 0 )$? If so, is the point a local maximum or a local minimum? Justify your answer.
Consider the differential equation $\frac { d y } { d x } = 2 y - 4 x$.
(a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through the point $( 0, 1 )$ and sketch the solution curve that passes through the point $( 0 , - 1 )$.
(b) Let $f$ be the function that satisfies the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with a step size of 0.1, to approximate $f ( 0.2 )$. Show the work that leads to your answer.
(c) Find the value of $b$ for which $y = 2 x + b$ is a solution to the given differential equation. Justify your answer.
(d) Let $g$ be the function that satisfies the given differential equation with the initial condition $g ( 0 ) = 0$. Does the graph of $g$ have a local extremum at the point $( 0, 0 )$? If so, is the point a local maximum or a local minimum? Justify your answer.