*The plot of $f$.* The graphical method for solving differential equations involves two different types of graphs. The first is a plot of the function defining the differential equation. Since our differential equation can be written as $u'(t)=f(u)$ for $f(u)=1-u^2/9$, we plot $f$, shown below.

Our continuous dynamical system (or differential equation) has one state variable $u$, which we could also write as $u(t)$, to emphasize its value depends on time. The variable $u$ lives on the horizontal axis (or $u$-axis) of above graph. Highlight the horizontal axis thickly to remind us this is where the action occurs.

At first, we'll consider the initial condition $u_0=-2$. Draw a point on the $u$-axis at the location $u=-2$. This point shows the initial state of the system, at time $t=0$.

You probably computed $f(-2)$ from the formula. Such precise information is overkill for this approach. Instead, use the graph of $f$ to estimate $f(-2)$. The graph indicates $f(-2)$ is positive; $u(t)$ must be increasing at $t=0$. For $t$ slightly larger than 0, $u(t)$ must become larger than $-2$. We'll indicate this change in two different ways on the two graphs.

On the plot of $f$, draw an arrow pointing right from the initial condition point you drew. $u(t)$ is moving rightward along the $u$ axis in that plot.

On the solution plot, start to draw the curve $u(t)$. Given that $u'(0)$ is positive, the curve begins with a positive slope. Draw a little bit of the curve $u(t)$ starting at $t=0$, making it slope upward. You can ignore the scale on the $t$ axis, so don't worry about the exact value of the slope.