Problem 1 - 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. $$ x^{\prime \prime}+9 \sin x=0 $$

Step-by-Step Solution

Answer

The system as plane autonomous is \( y'_1 = y_2 \), \( y'_2 = -9 \sin(y_1) \) with critical points at \( (n\pi, 0) \) for all integers \( n \).

Step 1: Rewrite the Second-order Differential Equation

We start with the given second-order differential equation: \[ x'' + 9 \sin x = 0. \] To convert it into a system of first-order differential equations, we set two new variables: \( y_1 = x \) and \( y_2 = x' \). Therefore, the second derivative becomes \( y_2' = x'' \).

Step 2: Express the System as First-order Equations

We can now express the original differential equation as a system by substituting the new variables. We have:1. \( y_1' = y_2 \) (since \( y_1 = x \) and \( y_2 = x' \)),2. \( y_2' = -9 \sin(y_1) \) (from the original equation \( x'' = -9 \sin x \)).Therefore, the plane autonomous system is:\[y'_1 = y_2,y'_2 = -9 \sin(y_1).\]

Step 3: Determine the Critical Points

Critical points occur where both derivatives in the system are zero. Thus, we set:1. \( y'_1 = y_2 = 0 \),2. \( y'_2 = -9 \sin(y_1) = 0 \).From \( y'_2 = 0 \), we get \( \sin(y_1) = 0 \), which gives \( y_1 = n\pi \) for integer \( n \). Since \( y_2 = 0 \), the critical points are \( (n\pi, 0) \) for all integers \( n \).

Key Concepts

Autonomous Systems

Autonomous systems in differential equations are significant because they are sets of equations that do not explicitly depend on the independent variable, usually time. This means that the equations describe how a system evolves over time solely based on its current state.
In mathematical terms, an autonomous system can be written in the form: \( \dot{y} = f(y) \), where \( y \) is a vector representing the system's state, and \( \dot{y} \) is the derivative of \( y \) with respect to time.
The beauty of autonomous systems is that they often model real-world phenomena, such as population growth or the motion of a pendulum, where the rate of change is dependent only on the current state and not the time of observation.

• They simplify analysis, since solutions do not explicitly rely on time.
• Phase plane analysis is often used to study them, visualizing trajectories in a state-space.

For the given exercise, converting the second-order differential equation into a plane autonomous system helps in identifying its critical points, leading us to understand its behavior over time.

Critical Points

Critical points in a differential equation system are essentially the states where the system does not change. These are the points where both first-order derivatives in the system are zero, implying a kind of equilibrium.
In our exercise, finding critical points involved setting the derivatives \( y'_1 \) and \( y'_2 \) to zero. This led to solutions where the pendulum returns to a state of equilibrium.

• The first condition \( y'_1 = y_2 = 0 \) implies no movement in the state variable.
• The second condition \( y'_2 = -9 \sin(y_1) = 0 \) results in several angles where this holds true.

These angles, specifically \( y_1 = n\pi \) for integer \( n \), indicate positions where a pendulum, for instance, would momentarily be at rest. It’s worth noting that not all critical points are stable; some may be saddle points or centers influencing the system's behavior.

First-order Differential Equations

First-order differential equations involve derivatives of only the first degree of a function. They are foundational because more complex equations can often be transformed into systems of first-order equations.
In the context of our exercise, transforming the original second-order equation into a system of two first-order equations made it easier to identify and analyze critical points.
The process of reducing a higher-order equation involves introducing new variables to represent the derivatives, as done here with \( y_1 = x \) and \( y_2 = x' \).

• Allows for simpler numerical solutions and analytical methods.
• Can be studied using graphical methods like direction fields.

Understanding first-order systems allows for deeper insights into systems that evolve based on their current state, making them crucial for many applications in science and engineering.

Second-order Differential Equations

Second-order differential equations are equations that include the second derivative of a function. They are particularly important in physics and engineering as they naturally describe many systems with inertia, like springs and pendulums.
For the exercise, the second-order equation \( x'' + 9 \sin x = 0 \) describes the motion of a pendulum under gravity, ignoring external forces like friction.
Converting a second-order equation to a first-order system is a common approach in solving, as it allows use of tools applicable to systems of first-order equations.

• Often model oscillatory and wave phenomena.
• Can describe systems with acceleration as a factor.
• Solvable using methods like the characteristic equation or numeric approaches like Runge-Kutta methods.

Mastery of second-order differential equations and methods to transform them provides the necessary groundwork to tackle many problems in dynamic systems and is essential for fields such as mechanical and electrical engineering.