Problem 43 - Calculus for Biology and Medicine

Question

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d y}{d x}=(y-2)(y+4)+h $$

Step-by-Step Solution

Answer

There are equilibria for \( h \leq 9 \). Stability requires analyzing sign changes in \( \frac{d y}{d x} \).

Step 1: Set the derivative to zero

To find the points of equilibrium, set the equation \( \frac{d y}{d x} = (y-2)(y+4) + h \) equal to zero: \ \[(y-2)(y+4) + h = 0.\]

Step 2: Solve for y in terms of h

Rearrange the equation to solve for \( y \): \[(y-2)(y+4) = -h.\] Expand the left side:\[y^2 + 2y - 8 = -h.\] Thus, \[y^2 + 2y + h - 8 = 0.\] This is a quadratic equation in \( y \).

Step 3: Determine the discriminant for equilibrium

For the quadratic equation \( y^2 + 2y + (h-8) = 0 \), determine the discriminant \( \Delta = b^2 - 4ac \). Here, \( a = 1 \), \( b = 2 \), and \( c = h-8 \). The discriminant is:\[\Delta = 2^2 - 4 \times 1 \times (h-8) = 4 - 4(h-8).\] Simplify the expression:\[\Delta = 4 - 4h + 32 = 36 - 4h.\] The existence of real solutions is determined by \( \Delta \geq 0 \).

Step 4: Conditions for real solutions

Solve \( 36 - 4h \geq 0 \) for \( h \):\[4h \leq 36,\] \[h \leq 9.\] Thus, the values of \( h \) for which there are equilibria are \( h \leq 9 \).

Step 5: Stability Analysis

Evaluate the stability by considering the sign of \( \frac{d y}{d x} \) around the equilibria: - For \( y^2 + 2y + (h-8) = 0 \), find roots (equilibria) \( y_1 \) and \( y_2 \). - Analyze \((y-2)(y+4)+h)\) around these points: - If \( \frac{d y}{d x} < 0 \) between roots, the equilibrium is stable. - If \( \frac{d y}{d x} > 0 \) outside roots, the equilibrium is unstable.

Key Concepts

Stability Analysis

In the context of differential equations, an equilibrium point is a value where the rate of change (derivative) of a function becomes zero. To determine the stability of these points, we examine how the function behaves when slightly perturbed. Stability analysis helps us understand whether disturbances grow or decay. This can tell us if equilibrium points are stable, unstable, or semi-stable.
To assess stability:

• If a slight increase in the function leads to a decrease back toward equilibrium, the point is stable.
• If a slight decrease also returns the function to equilibrium, stability is confirmed.
• If any such perturbation leads to the function moving away from equilibrium, it's unstable.

Determining stability involves checking the sign of the derivative around equilibrium. A negative derivative between roots indicates stability, while a positive derivative suggests instability.

Quadratic Equation

A quadratic equation is any equation that can be written in the form: \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants. In stability analysis, finding equilibrium often leads to solving a quadratic equation, as demonstrated in the earlier steps.

• To find the solutions (roots) of the quadratic equation, the standard quadratic formula is used: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• The roots \( y_1 \) and \( y_2 \) identify potential equilibrium points.
• Knowing the roots can help in determining the nature of stability by checking changes in the sign of the derivative around these roots.

Quadratic equations are central in mathematical modeling, where they simplify complex phenomena into understandable parts.

Discriminant Analysis

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( \Delta = b^2 - 4ac \). This important value reveals crucial information about the nature of the equation's roots.

• If \( \Delta > 0 \), the equation has two distinct real roots. This means there are two equilibria to consider, and each can be analyzed for stability.
• If \( \Delta = 0 \), there is exactly one real root, indicating a repeated root and potentially semi-stable equilibrium.
• If \( \Delta < 0 \), no real roots exist, implying no equilibrium points.

The discriminant helps in setting the conditions, such as \( h \leq 9 \) in our exercise, to ensure real equilibria exist. Analyzing the discriminant streamlines our understanding of the behavior of dynamic systems modeled by quadratic equations.