Even before you learn techniques for solving differential equations, you may be able to analyze equations qualitatively . As an example, look at the nonlinear equation $$\frac{dy}{dt}=y\!\left(y-1\right)$$ You are going to analyze the solutions, $y$, of this equation without actually finding them. You will be asked to sketch three solutions of the differential equation on the graph below based on qualitative information from the differential equation.

In what follows, picture the $t$-axis running horizontally and the $y$ axis running vertically. There is no scale on the $t$ axis but imagine it is large enough to display the behavior of the solutions as $t$ approaches $\pm \infty$.

a) For what values of $y$ is the graph of $y$ as a function of $t$ increasing? Use interval notation for your answer.

b) For what values of $y$ is the graph of $y$ concave up?
For what values of $y$ is it concave down? (Help with interval notation.)
What information do you need to answer a question about concavity? Remember that $y$ is an implicit function of $t$. [How to enter answer]

Parts c),d),e) of this question ask you to modify the blue, red, and green curves in the plot below to make them represent graphs of particular solutions of the differential equation.

To modify the blue curve, click the "blue curve" button below the plot to expose the blue points and tangents. Solid blue points lie on the curve. With your mouse click and hold each solid blue point, and move it into a better position. If the solution curve crosses an edge of the viewing region then the solid point should be very near the edge, left or right, top or bottom. Improve the shape of the curve between the solid points by moving the open points that lie on the dashed tangents. Experiment to see how the shape changes. Small errors in concavity are hard to see -- you may have to fiddle with the tangents to get the right concavity.

Modify the red or green curve in a similar way, after clicking the corresponding button to expose its points and tangents. I recommend moving the solid points into good positions first, then move the open points to improve the shape between the solid points.

$$\frac{dy}{dt}=y\!\left(y-1\right).$$

Display points and tangents on:
blue curve red curve green curve no curve

c) BLUE: Use the information found in parts (a) and (b) to modify the blue curve to make it represent the graph of a solution $y=y(t)$ with initial condition $y(0)=0.5$.
What is the long-term behavior of this solution $y(t)$? That is, what is $\lim_{t\to\infty}y(t)$?

d) RED: Next, use the information found in parts (a) and (b) to modify the red curve to make it represent the graph of a solution $y=y(t)$ with initial condition $y(0)=2$.
With this initial condition, what is the long-term behavior of $y(t)$? That is, what is $\lim_{t\to\infty}y(t)$?

e) GREEN: Finally, based on what you see in the original differential equation, modify the green curve to make it represent the graph of a solution $y=y(t)$ with initial condidtion $y(0)=1$.

f) If $y(t)$ represents the population of some animal species, and if units on the $y-$axis are in thousands, interpret the results of (c), (d) and (e).
The solution to part (c) (sketched in blue ) represents: ? extinction equilibrium explosive growth

The solution to part (d) (sketched in red ) represents: ? extinction equilibrium explosive growth

The solution to part (e) (sketched in green ) represents: ? extinction equilibrium explosive growth

Adapted from A Modern Introduction to Differential Equations, 2nd Ed., Henry J. Ricardo, 2009.