Match the differential equation with its direction field.y' = 2-y-41.0Q,51.5 1-1.0 I +0,5\1$ Q51.0O-21.00,5-1,5- -1.O - -0.5 -0.5 -1.0 - -15ニ6.51.5-O-2Give reasons for your answer.O y' = 2 - y = 0 on the line y = -x + 1/2, and y' = -1 on the line y = -x.O The slopes at each point are Independent of x, so the slopes are the same along each line parallel to the x-axis. Notethat for y = 2, y' = 0.O y' = 2 - y = 0 on the lines x = 0 and y = 2O The slopes at each point are Independent of y, so the slopes are the same along each line parallel to the y-axis. Notethat for y = 2, y' = 0.O y' = 2 - y = 0 on the lines x = 0 and y = 0, and y' > 0 for o < x < n/2, 0

The given differential equation is, y'=2-y. Note that, the derivative is independent of x. Also, y'=0 implies, 2-y=0y=2 That is, the derivative is 0 at y=2. Thus, the graph of the slope field near y=2 will be a parallel to the x-axis. From the given graphs, the second and third graph do not show slope field lines at y=2 as parallel to x- axis. Eliminate the second and third graphs. So, only the first and second graph has slope field parallel to x-axis at y=2. Now, check for the increasing and decreasing property. Explore the regions where graph has increasing and decreasing slope. Increasing y'=2-y>0 implies, 2-y>02>y Decreasing y'=2-y<0 implies, 2-y<02<y That is, the derivative is greater than 0 for y<2 and less than 0 for y>2. So, the graph should be increasing for all values below y=2 and decreasing for all above y=2. Consider the first graph, the slope field line increases in the first quadrant and then decreases in the second quadrant. Since the first graph is not increasing for all values below y=2 and decreasing for all values above y=2, it does not represent the given differential equation. Checking the fourth graph, it does increase for all values below y=2 and decrease for all above y=2. Thus, the graph that represents the given differential equation y'=2-y is the fourth one. From the above figure, it is observed that the slopes are same along each line parallel to the x-axis. The explanation is: The slopes at each point are independent of x. The slopes are same along each line parallel to the x-axis. For y=2, the derivative y'=0. Therefore, combining all the above points, the correct option to the reasoning is the second one.