Answer
The correct differential equation is: $~~C. ~~y' = 1-2xy$
Work Step by Step
Let's consider differential equation $A$:
$y' = xy+1$
According to this equation, the slope of the graph $y'$ must be positive when both $x$ and $y$ are positive. However, on the graph, we can see that the slope of the graph is negative for some parts of the graph where both $x$ and $y$ are positive. Therefore, this is not the correct differential equation.
Let's consider differential equation $B$:
$y' = -2xy$
According to this equation, the slope of the graph $y'$ must be 0 when $x=0$. However, on the graph, we can see that the slope of the graph is positive when $x=0$. Therefore, this is not the correct differential equation.
Let's consider differential equation $C$:
$y' = 1-2xy$
Since the other two equations have been eliminated, by the process of elimination, this differential equation must be correct. Also, since we can see that the slope along the graph seems to match up well with this differential equation, this is the correct differential equation.
