Question

What condition on \(\lambda\) will ensure that the straight line solution \(\mathbf{x}(t)=e^{\lambda t} \mathbf{v}\) moves toward the equilibrium point at the origin as time increases? What condition ensures that the straight line solution will move away from the equilibrium point as time increases?

Short answer

\(\lambda < 0\) ensures moving toward the origin; \(\lambda > 0\) ensures moving away.

Step-by-step solution

Understanding the Problem

We need to find the condition on \( \lambda \) such that the solution \( \mathbf{x}(t) = e^{\lambda t} \mathbf{v} \) approaches or moves away from the equilibrium point at the origin as \( t \to \infty \). The direction of \( \mathbf{x}(t) \) depends on the sign of \( \lambda \).

Condition for Moving Toward the Origin

For \( \mathbf{x}(t) = e^{\lambda t} \mathbf{v} \) to move towards the origin as \( t \) increases, the exponential term \( e^{\lambda t} \) must approach zero. This happens when \( \lambda < 0 \) because the exponential function decays to zero for negative arguments as \( t \to \infty \). Therefore, the condition for moving towards the origin is \( \lambda < 0 \).

Condition for Moving Away from the Origin

Conversely, \( \mathbf{x}(t) \) will move away from the origin if \( e^{\lambda t} \) grows as \( t \) increases. This occurs when \( \lambda > 0 \) since the exponential function increases to infinity for positive arguments as \( t \to \infty \). Thus, the condition for moving away from the origin is \( \lambda > 0 \).

Special Case Discussion

If \( \lambda = 0 \), the expression \( e^{\lambda t} = e^0 = 1 \) becomes constant over time. Therefore, \( \mathbf{x}(t) \) remains at a fixed distance from the origin, neither moving closer nor further away.

Key concepts

Equilibrium Points

In ordinary differential equations, equilibrium points play a crucial role in understanding the behavior of systems. An equilibrium point is where the solution to a differential equation remains constant over time. In simpler terms, it's like finding a spot where nothing is moving or changing.

To determine an equilibrium point, we set the derivative of the function to zero. This helps us explore whether the function stays still when it gets there. For example, when dealing with systems like a pendulum or population dynamics, equilibrium points tell us whether the system will stay stable over time or if it will change.

• If a system tends to move towards an equilibrium point, it's called stable.
• If it moves away, it's unstable.

Understanding equilibrium points helps predict how systems will behave in the future, especially in engineering and physics. By examining these points, engineers can ensure systems operate safely without unexpected surprises.

Exponential Solutions

Exponential solutions are a core concept in differential equations, often of the form \( \mathbf{x}(t) = e^{\lambda t} \mathbf{v} \).Here, \( e^{\lambda t} \) acts like a magnifying or shrinking glass, affecting how the system behaves over time.

The expression \( e^{\lambda t} \) depends on \( \lambda \), which determines whether it will grow, shrink, or remain constant.

• If \( \lambda < 0 \), \( e^{\lambda t} \) approaches zero as time increases, indicating the solution shrinks.
• If \( \lambda > 0 \), \( e^{\lambda t} \) increases, meaning the solution grows over time.
• When \( \lambda = 0 \), \( e^{\lambda t} = 1 \) remains steady, so the solution doesn't change.

These behaviors are critical in systems governed by ordinary differential equations, such as circuits and population models. Understanding exponential behavior allows for predicting how a solution will evolve or stabilize.

Eigenvalues

Eigenvalues are fascinating and essential in understanding dynamic systems, especially within the context of linear algebra and differential equations. They represent special numbers associated with matrices or linear transformations, showing how a vector's direction and magnitude are affected.

In the context of differential equations, eigenvalues tell us about the stability of systems. Consider the equation \( \mathbf{x}(t) = e^{\lambda t} \mathbf{v} \), where \( \lambda \) represents eigenvalues. The sign of \( \lambda \) directly impacts the solution's behavior over time.

• \( \lambda < 0 \): The system is stable, as solutions move towards equilibrium (e.g., dampening oscillations).
• \( \lambda > 0 \): The system is unstable, indicating growth away from equilibrium (e.g., uncontrolled population growth).
• \( \lambda = 0 \): A neutral state where changes don't affect the solution's growth or decay.

Eigenvalues are instrumental in engineering, physics, and economics for designing stable systems and anticipating changes. By analyzing eigenvalues, we can understand whether systems will balance themselves or veer unpredictably.