Problem 8 - Advanced Engineering Mathematics

Question

The corresponding plane autonomous system is $$x^{\prime}=y, \quad y^{\prime}=-x+\left(\frac{1}{2}+3 y^{2}\right) y-x^{2}$$ and so \(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}=0+\frac{1}{2}+9 y^{2} > 0 .\) Therefore there are no periodic solutions by Theorem 11.5.

Step-by-Step Solution

Answer

No periodic solutions exist due to the positivity of \(\frac{\partial Q}{\partial y}\).

Step 1: Understand the system

The given dynamic system is defined by two differential equations: \(x' = y\) and \(y' = -x + \left(\frac{1}{2} + 3y^2\right)y - x^2\). These describe how \(x\) and \(y\) change with respect to time.

Step 2: Partial derivatives calculation

To determine the nature of solutions, we calculate the partial derivatives of \(P\) and \(Q\). In this case, \(\frac{\partial P}{\partial x} = 0\) and \(\frac{\partial Q}{\partial y} = \frac{1}{2} + 9y^2\).

Step 3: Analyzing the expression

Combine these partial derivatives to analyze their sum, \(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0 + \frac{1}{2} + 9y^2\). This expression is always positive since \(y^2\) is non-negative.

Step 4: Apply Theorem 11.5

Theorem 11.5 states that if \(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} > 0\), then the system has no periodic solutions. Since the expression is always positive, periodic solutions do not exist.

Key Concepts

Differential Equations

Differential equations are mathematical equations that describe how a quantity changes over time. They are essential in modeling real-world phenomena where change is involved, such as population growth, heat transfer, or motion dynamics. In the context of autonomous systems, differential equations can describe how variables evolve without explicit dependence on time.

In the provided exercise, we deal with the system of differential equations given by:

• \( x' = y \)
• \( y' = -x + \left(\frac{1}{2} + 3y^2\right) y - x^2 \)

These equations define how the variables \( x \) and \( y \) change with respect to another variable, often time. The notation \( x' \) and \( y' \) signifies the derivatives of \( x \) and \( y \), respectively, indicating rates of change.

Understanding these equations requires determining how \( x \) and \( y \) will behave. They form what is known as an autonomous system because they do not include the independent variable (usually time) explicitly. Such systems are insightful in exploring steady states or equilibrium points, where \( x' \) and \( y' \) are simultaneously zero, indicating no change over time.

Partial Derivatives

Partial derivatives are used in functions of multiple variables to capture how the function changes with respect to one variable, holding others constant. They are integral in analyzing equations that involve more than one variable, like our system of differential equations.

To better understand our system, we calculate the partial derivatives of functions \( P \) and \( Q \). For the given exercise, these are derived from:

• \( \frac{\partial P}{\partial x} = 0 \)
• \( \frac{\partial Q}{\partial y} = \frac{1}{2} + 9y^2 \)

Here, \( P \) and \( Q \) reflect the right-hand sides of our differential equations, typically modeled as \( P(x, y) \) and \( Q(x, y) \).

Calculating these derivatives helps clarify how shifts in one variable might influence the output altogether. In effect, partial derivatives serve as building blocks in constructing a broader understanding of the system's behavior. Such calculations are crucial when evaluating conditions like stability in dynamical systems, often leading to criteria for determining equilibrium or periodic behavior.

Theorem 11.5

Theorem 11.5 offers a tool to assess the existence of periodic solutions in dynamical systems. It relies on the sum of partial derivatives, \( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \).

The mechanism of Theorem 11.5 indicates that if this sum is always greater than zero, then the system has no periodic solutions. This is often employed in proving the non-existence of cycles or recurring states in autonomous systems, ensuring their dynamics do not repeat over time.

In our specific exercise, the calculation yields:

• \( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0 + \frac{1}{2} + 9y^2 \)

The term \( \frac{1}{2} + 9y^2 \) is strictly positive since \( y^2 \geq 0 \). Therefore, applying Theorem 11.5 with this expression greater than zero confirms the absence of periodic solutions. This theorem provides an elegant solution to a complex problem, highlighting why such mathematical tools are valued in the study of differential equations and dynamical systems.