Problem 17 - A First Course in Differential Equations with Modeling Applications

Question

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &D x=y\\\ &\begin{array}{l} D y=z \\ D z=x \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

The solution forms depend on oscillatory functions like sine and cosine.

Step 1: Write Down the Given System

We are given the system of differential equations: \[ \begin{aligned} \ Dx &= y \ Dy &= z \ Dz &= x \end{aligned} \] Our goal is to solve this system through systematic elimination.

Step 2: Differentiate Each Equation

We'll differentiate each equation with respect to \( t \):- Differentiate \( Dx = y \), we get \( D^2x = Dy \).- Differentiate \( Dy = z \), we get \( D^2y = Dz \).- Differentiate \( Dz = x \), we get \( D^2z = Dx \).

Step 3: Substitute Back into Original Equations

We substitute these new equations back into our system to eliminate variables. For instance:- Substitute \( Dy = z \) into \( D^2x = Dy \) gives \( D^2x = z \).- Substitute \( Dz = x \) into \( D^2y = Dz \) gives \( D^2y = x \).- Substitute \( Dx = y \) into \( D^2z = Dx \) gives \( D^2z = y \).

Step 4: Formulate the Final Set of Differential Equations

We now rewrite the system using the substitutions:\[ \begin{aligned} D^2x &= z \D^2y &= x \D^2z &= y \end{aligned} \]We've reduced the problem to solving these second-order differential equations.

Step 5: Identify and Solve for Eigenvalues and Eigenvectors

Lump the substituted system into matrix form, \[ D^2 \begin{bmatrix} x \ y \ z \end{bmatrix}= \begin{bmatrix} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} \] Solve for eigenvalues and eigenvectors of the coefficient matrix, yielding solutions periodic in nature.

Step 6: Formulate the General Solution

Using the eigenvalues and eigenvectors, construct the general solution for each variable:\[ x(t), y(t), z(t) \] satisfies coupled sinusoidal and exponential forms such as:\[ \begin{bmatrix} x(t) \ y(t) \ z(t) \end{bmatrix} = c_1 \begin{bmatrix} \cos(t) \ \sin(t) \ \cos(t) \end{bmatrix} + c_2 \begin{bmatrix} \sin(t) \ \cos(t) \ \sin(t) \end{bmatrix} + c_3 \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \] where \( c_1, c_2, c_3 \) are constants determined by initial conditions.

Key Concepts

Systematic Elimination

Solving systems of differential equations can sometimes appear challenging, but systematic elimination offers a structured approach to simplify these problems. This method involves reducing the number of variables by differentiating and substituting equations until a simpler form is achieved. Using systematic elimination begins by differentiating each equation in the system with respect to time, denoted as \( t \).

For example, consider the system:

• \( Dx = y \)
• \( Dy = z \)
• \( Dz = x \)

When differentiated, each equation leads to a new form that involves fewer dependent variables. Substitutions are then made to replace terms with their differentiated counterparts, gradually eliminating the original variables. This process continues until the equations all feature one independent variable in terms of itself, like the second-order differential equations for \( x, y, \) and \( z \) in terms of \( z, x,\) and \( y \) respectively, effectively decoupling the system.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental in understanding linear transformations and their effect on vector spaces. They provide a way to solve systems of linear equations, such as those found in differential equations. In matrix notation, the given system of differential equations can be rewritten, allowing us to use these mathematical tools.

The coefficient matrix for our system was:\[\begin{bmatrix} 0 & 0 & 1 \1 & 0 & 0 \0 & 1 & 0 \end{bmatrix}\]Finding the eigenvalues of this matrix involves solving the characteristic equation, typically in the form \( |A - \lambda I| = 0 \), where \( A \) is the matrix, \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix.

Once the eigenvalues are determined, eigenvectors are obtained by substituting each eigenvalue back into the equation \( (A - \lambda I)\vec{v} = 0 \), solving for \( \vec{v} \). These offer insights into periodic or exponential behavior in the solutions, critical when these solutions describe physical systems such as oscillations.

Second-Order Differential Equations

A second-order differential equation involves derivatives up to the second degree, often appearing in the study of dynamic systems. The system of equations we encountered translates into a set of second-order differential equations after differentiating and substituting:

• \( D^2x = z \)
• \( D^2y = x \)
• \( D^2z = y \)

Each equation expresses the second derivative of one variable in terms of another. Solving these equations allows us to explore how the variables evolve over time, which is crucial in understanding complex systems like the motion of particles or circuits.

In general, solutions to these equations can be expressed in sinusoidal and exponential terms when solved using techniques such as undetermined coefficients or variation of parameters. Understanding the interplay of second derivatives helps in modeling and predicting the future states of dynamic systems.