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A phase line is a simple visual tool that helps us understand the stability of equilibrium points in differential equations. It shows the direction of movement of solution trajectories in relation to these points. Picture a horizontal line representing the values of the variable of interest. In our example, this would be values of \( y \).
Consider when \( y < 1 \): the derivative \( y' < 0 \), indicating that any initial value below 1 will move toward \( y = 1 \). Mark this by drawing arrows pointing toward one from the left. When \( y > 1 \), \( y' > 0 \); values above 1 will also move back toward the equilibrium at \( y = 1 \). Here, you would draw arrows pointing toward one from the right. This balanced movement from both directions shows that \( y = 1 \) is a stable equilibrium point.

The signs of a derivative give us clues about the behavior of a function's graph. Let's start with the first derivative, \( y' = y - \sqrt{y} \). If \( y' < 0 \), the graph of the function is decreasing; if \( y' > 0 \), it is increasing.
To find the stability around the equilibrium \( y = 1 \), check the derivative signs for values just below and just above. For \( y = 0.5 \), calculate \( y' = 0.5 - \sqrt{0.5} \approx -0.207 \), so the graph decreases towards zero. For \( y = 1.5 \), \( y' = 1.5 - \sqrt{1.5} \approx 0.776 \); the graph increases. Therefore, we see stability since trajectories move back to \( y = 1 \) regardless of starting position.
The second derivative \( y'' \) tells us about concavity. For \( y = 1 \), \( y'' = 1 - \frac{1}{2\sqrt{1}} = 0.5 \) implies it is concave up around this point, confirming \( y = 1 \) is a minimum.

Concavity explains whether a curve is bending upwards or downwards and is found using the second derivative. A function is concave up when \( y'' > 0 \) and concave down when \( y'' < 0 \).
In this example, we derived \( y'' = 1 - \frac{1}{2\sqrt{y}} \). For values close to the stable equilibrium \( y = 1 \), substitute to get \( y'' = 1 - 0.5 = 0.5 \), indicating the graph curves upwards, or is concave up.
This tells us more context for sketching solution curves: when approaching the stability point at \( y = 1 \), the graph turns upward, creating a U-shape. This clue complements the information from the first derivative and phase line analysis.

Sketching solution curves from our analysis rounds out our understanding of a differential equation's behavior. With phase line, derivative sign, and concavity in mind, plotting these curves becomes clearer.
Since \( y' \) shows stability around \( y = 1 \), the solutions are attracted to this point from both sides. Points where \( y < 1 \) fall towards one, creating increasingly shallower angles as the graph slides up to \( y = 1 \). Points starting at \( y > 1 \) head down towards the line as well.
The second derivative hints solutions will keep a concave paradise, bending upward as they near \( y = 1 \). This analysis outlines the overall shape as a soothing curve gently bending upward approaching the equilibrium, which matches our expectations for stability and concavity.