Problem 4 - Advanced Engineering Mathematics

Question

In Problems, write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+4 \frac{x}{1+x^{2}}+2 x^{\prime}=0 $$

Step-by-Step Solution

Answer

The critical point is (0, 0).

Step 1: Recognize the Differential Equation Format

Start by identifying the form of the given second-order differential equation: \(x'' + 4\frac{x}{1+x^2} + 2x' = 0\). Notice it's a second-order with respect to \(x\).

Step 2: Transform into Two First-order Equations

Introduce a substitution to reduce the second-order equation into two first-order equations. Set \(y = x'\), leading to the system:\[\begin{align*}x' &= y, \y' &= -4\frac{x}{1+x^2} - 2y.\end{align*}\]

Step 3: Write the System in Standard Planar Form

The system of first-order equations can be written in the form:\[\begin{align*}\frac{dx}{dt} &= y, \\frac{dy}{dt} &= -4\frac{x}{1+x^2} - 2y.\end{align*}\]This is now in the standard plane autonomous system format.

Step 4: Find Critical Points

Set both derivatives equal to zero to find the critical points. Solving \(y = 0\) and \(-4\frac{x}{1+x^2} - 2y = 0\), we find:- From \(y = 0\), substitute into \(-4\frac{x}{1+x^2} = 0\), this simplifies to \(x = 0\).- Thus, the only critical point is \((x, y) = (0, 0)\).

Key Concepts

Second-order Differential Equations

A second-order differential equation involves the second derivative of a function. These equations are key in describing systems where the rate of change of momentum or velocity varies, such as in harmonic oscillators. Our exercise presents the equation: \[ x'' + 4\frac{x}{1+x^2} + 2x' = 0 \] Here, \(x''\) signifies the second derivative of \(x\) concerning time \(t\). The goal is to understand how such an equation can model dynamic systems. We first convert it into an equivalent system of first-order differential equations. This helps in analyzing and solving complex systems by simplifying them into more manageable chunks. By breaking it down, we can more easily study system behavior and stability. Remember, second-order linear equations have constant coefficients and solutions exhibit predictable behaviors when compared to nonlinear forms as in our case.

Autonomous Systems

An autonomous system in differential equations is one where the equations do not explicitly depend on the independent variable, usually time \(t\). Instead, it relies on the functions themselves. Our exercise involves transforming a second-order differential equation into a system of first-order equations:

• \(x' = y\)
• \(y' = -4\frac{x}{1+x^2} - 2y\)

This transformation is crucial as it helps analyze the system behavior independently of time, emphasizing state over time-dependency. Autonomous systems are prevalent in modeling steady-state behaviors where the focus is on equilibrium or stability rather than temporal evolution. The power of such systems lies in their ability to illustrate critical points and their dynamics without being anchored to time sequences.

Critical Points

Critical points, also known as equilibrium points, are where the state variables of a system do not change over time. In our context, they are the solutions to the system of equations when both derivatives are zero:

• \(y = 0\)
• \(-4\frac{x}{1+x^2} = 0\)

By solving these, we find that \(x = 0\) and \(y = 0\), meaning our system's only critical point is \((0, 0)\). Identifying such points is important because they help us discern system behaviors—whether a system returns to a stable state after a disturbance or moves away towards chaos. Analyzing the stability around critical points typically involves evaluating the eigenvalues of the Jacobian matrix at these points.

Plane Systems

Plane systems refer to the representation of differential equations in a two-dimensional space, generally in terms of \(x\) and \(y\). In such a setup, an autonomous system can be visualized as vector fields on the \(xy\)-plane:

• \(\frac{dx}{dt} = y\)
• \(\frac{dy}{dt} = -4\frac{x}{1+x^2} - 2y\)

Representing equations this way makes it easier to visualize solutions as trajectories in the plane, allowing us to observe patterns such as spirals, nodes, and saddle points. These graphical representations provide insights into a system's phase portrait, showing how initial conditions evolve over time. Plane systems simplify examination of complex dynamics and offer a valuable way to predict long-term behaviors and system responses to initial parameters.