Problem 50 - Calculus for Biology and Medicine

Question

Determine whether the equilibrium at \(x=0\) is stable, unstable, or semi- stable. $$ \frac{d x}{d t}=-x^{5} $$

Step-by-Step Solution

Answer

The equilibrium at \( x=0 \) is stable.

Step 1: Determine the Equilibrium Points

The first step is to determine the equilibrium points of the differential equation. These occur where \( \frac{dx}{dt} = 0 \). For the given function, we have \( \frac{dx}{dt} = -x^5 = 0 \). This implies that \( x = 0 \) is the only equilibrium point.

Step 2: Analyze the Sign of the Derivative

Examine the sign of \( \frac{dx}{dt} = -x^5 \) around the equilibrium point \( x=0 \). For \( x > 0 \), \( -x^5 < 0 \) (negative), and for \( x < 0 \), \( -x^5 > 0 \) (positive).

Step 3: Determine the Type of Stability

To determine the type of stability, observe the behavior around the equilibrium point. For \( x > 0 \), \( x \) decreases because \( \frac{dx}{dt} \) is negative, and for \( x < 0 \), \( x \) increases because \( \frac{dx}{dt} \) is positive. This indicates that the equilibrium at \( x=0 \) is stable because solutions approach \( x = 0 \) from both sides.

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve an unknown function and its derivatives. They are essential in modeling various real-world phenomena such as population growth, heat flow, and motion.

In general, a differential equation relates some function with its rate of change. For instance, in our exercise, we have \[ \frac{dx}{dt} = -x^{5} \]This equation indicates how the variable \( x \) changes with respect to time \( t \). The power of differential equations lies in their ability to describe how dynamic systems evolve over time, adjusting to conditions specified by the equation's parameters.

Equilibrium Point

An equilibrium point for a differential equation is a point where the rate of change of the variable is zero. In simpler terms, it's a point where the solution of the differential equation doesn't change over time.

For the equation: \[ \frac{dx}{dt} = -x^5 \] We identify equilibrium by setting the derivative equal to zero: \( -x^5 = 0 \). This simplifies to \( x = 0 \), marking \( x = 0 \) as the equilibrium point.

• Equilibrium points are important because they represent steady states in a system.
• Once the system reaches an equilibrium, it will remain there unless disturbed.

Thus, identifying and analyzing equilibrium points is a vital step in understanding the behavior of a system governed by differential equations.

Sign Analysis

Sign analysis helps determine the type of stability of an equilibrium point by examining how the signs of the derivative change around the point. For our differential equation \[ \frac{dx}{dt} = -x^5 \], we assess this by observing:

• For \( x > 0 \), the derivative \( -x^5 \) is negative, indicating \( x \) will decrease as time progresses, moving towards \( x = 0 \).
• For \( x < 0 \), the derivative \( -x^5 \) is positive, so \( x \) increases over time, also heading towards \( x = 0 \).

This behavior suggests that small deviations from \( x = 0 \) will decay over time, confirming that \( x = 0 \) is a stable equilibrium point. Thus, sign analysis is crucial for determining whether an equilibrium is stable, avoiding excessive fluctuations or instability.