Consider the function $\frac{dy}{dx}=x^2+y-1$. Sketch isoclines for $k=0, 2, 4$ on a grid for $x, \, y \in [-5, 5].$ Hence, construct a slope field and sketch the solution curve satisfying $y(0)=1$.
I equated the derivative to the values of $k$ and obtained the following values for the isoclines:
$$y=-x^2+1$$ $$y=-x^2+3$$ $$y=-x^2+5$$
I graphed the isoclines (and the slope field) using Geogebra:
They indeed connect points having the same slope of the parent function.
But the question requires me to graph the isoclines and hence create the slope field (I wouldn't have Geogebra in class :) ). So assuming that the only thing we have on the graph above are the parabolas (and no slope field), how could we create the slope field from the isoclines alone and hence find the solution satisfying the conditions?
The answer, according to my textbook, is this function:
