Problem 2 - Advanced Engineering Mathematics

Question

Write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. \(x^{\prime \prime}+\left(x^{\prime}\right)^{2}+2 x=0\)

Step-by-Step Solution

Answer

Critical point: \((0, 0)\).

Step 1: Rewrite the Differential Equation

We begin with the given nonlinear second-order differential equation: \[ x'' + (x')^2 + 2x = 0 \]Our task is to convert it into a system of first-order differential equations. We let \( y = x' \). This substitution helps to rewrite the given equation as a system of first-order equations.

Step 2: Establish the System of Equations

Using our substitution \( y = x' \), we express the second derivative \( x'' \) in terms of \( y \) as follows:\[ x'' = y' \]. Therefore, the original equation becomes:\[ y' + y^2 + 2x = 0 \].Now we have the system of first-order differential equations:1. \( x' = y \)2. \( y' = -y^2 - 2x \)

Step 3: Identify the Autonomous System

The system of equations: \( x' = y \) and \( y' = -y^2 - 2x \) is an autonomous system because each equation depends only on the variables \( x \) and \( y \), and not explicitly on \( t \) or any other independent variable.

Step 4: Find Critical Points

To find the critical points of the system, set \( x' = 0 \) and \( y' = 0 \):1. \( x' = y = 0 \)2. \( y' = -y^2 - 2x = 0 \). Since \( y = 0 \), substituting into the second equation gives:\[ -0^2 - 2x = 0 \rightarrow x = 0 \]. Thus, the system has a critical point at \((x, y) = (0, 0)\).

Key Concepts

Autonomous Systems

An autonomous system is a type of dynamic system in which the change of state variables is determined entirely by the state itself and does not explicitly depend on time or any external factors. This makes autonomous systems a key area of study in solving differential equations, especially when analyzing stability and system behavior over time.

In our exercise, we converted the given second-order differential equation into a first-order system where the derivatives, expressed as functions of the state variables, were independent of time. Specifically, we dealt with the following system:

• \(x' = y\)
• \(y' = -y^2 - 2x\)

Because neither equation includes an external time variable, it is autonomous.

Key advantages of autonomous systems include:

• Simplicity. The behavior of the system depends only on its current state.
• Easier analysis of stability and equilibrium due to predictable behavior.

Understanding autonomous systems is fundamental when working with differential equations that describe natural phenomena.

Critical Points

Critical points are essentially the equilibrium points of a system of differential equations where the time derivatives (or the rates of change) of all variables become zero. These points help in understanding the long-term behavior of dynamic systems because they mark where the system can potentially stabilize.

In the context of the exercise, identifying the critical points required setting both \(x' = 0\) and \(y' = 0\). From our system of first-order differential equations:

• \(x' = y = 0\)
• \(y' = -y^2 - 2x = 0\)

Solving these equations, we found that the only critical point of the system is \((x, y) = (0, 0)\).

To further analyze these critical points, one could study their stability using linearization techniques, such as the Jacobian matrix. Such analysis reveals whether small perturbations will decay or amplify, informing us if the system will return to equilibrium or diverge from it.

First-Order Differential Equations

First-order differential equations involve the first derivative of a function and provide a simpler framework to analyze complex systems described by higher-order equations. Converting higher-order equations into a system of first-order equations is a common technique used to simplify the process of finding solutions.

In this exercise, the substitution \(y = x'\) allowed us to rewrite the second-order differential equation \(x'' + (x')^2 + 2x = 0\) as the system:

• \(x' = y\)
• \(y' = -y^2 - 2x\)

This transformation is powerful as it enables us to use the wealth of techniques developed for first-order systems, including numerical solutions and phase plane analysis.

Important aspects to remember about first-order differential equations include:

• Simplicity, allowing ease of computation and visualization in phase space.
• Their ability to describe a wide range of physical systems, from simple circuits to complex biological models.

Mastering first-order differential equations is crucial for anyone looking to solve practical problems in engineering or physics.