Problem 17 - Advanced Engineering Mathematics

Question

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &D x=y \\ &D y=z \\ &D z=x \end{aligned} $$

Step-by-Step Solution

Answer

The solution involves finding expressions for \(x(t), y(t), z(t)\) by solving the third-order differential equation derived from the system.

Step 1: Differentiate the first equation

Differentiate the first equation with respect to time:\[ D^2 x = D y \]This yields the second derivative of \(x\) in terms of \(y\).

Step 2: Substitute from the second equation

Substitute the expression for \(D y\) from the second differential equation:\[ D^2 x = z \]

Step 3: Differentiate the second equation

Differentiate the second equation with respect to time:\[ D^2 y = D z \]This helps relate \(y\) with \(z\) through their derivatives.

Step 4: Substitute from the third equation

Substitute the expression for \(D z\) from the third differential equation:\[ D^2 y = x \]

Step 5: Differentiate the third equation

Differentiate the third equation with respect to time:\[ D^2 z = D x \]This yields the second derivative of \(z\) in terms of \(x\).

Step 6: Substitute from the first equation

Substitute the expression for \(D x\) from the first differential equation:\[ D^2 z = y \]

Step 7: Form a single differential equation

Notice that we have now: \[ D^2 x = z,\quad D^2 y = x,\quad D^2 z = y \]Differentiate \(D^2 x = z\) again resulting in \(D^3 x = D z = x\). Therefore, the homogeneous third-order differential equation that each variable satisfies is:\[ D^3 x = x \]

Step 8: Solve the third-order differential equation

The third-order differential equation \(D^3 x - x = 0\) has solutions of the form:\[ x(t) = c_1 e^t + c_2 e^{(-1/2)t} \cos\left(\frac{\sqrt{3}}{2} t\right) + c_3 e^{(-1/2)t} \sin\left(\frac{\sqrt{3}}{2} t\right) \]Given its characteristic equation roots: \(1, -\frac{1}{2} \pm i\frac{\sqrt{3}}{2}\).

Step 9: Write solutions for y(t) and z(t)

Apply the same solution format to \(y\) and \(z\) considering cyclic permutations of initial conditions:- \( y(t) = c_1 e^t + c_2' e^{(-1/2)t} \cos\left(\frac{\sqrt{3}}{2} t\right) + c_3' e^{(-1/2)t} \sin\left(\frac{\sqrt{3}}{2} t\right) \)- \( z(t) = c_1 e^t + c_2'' e^{(-1/2)t} \cos\left(\frac{\sqrt{3}}{2} t\right) + c_3'' e^{(-1/2)t} \sin\left(\frac{\sqrt{3}}{2} t\right) \)

Key Concepts

Systematic Elimination

Systematic elimination is a process used to simplify and solve systems of differential equations. Think of it as a method to reduce these systems into manageable individual equations. The goal is to express a complex system in terms of a single variable or function and then find a solution for that variable.

In our original exercise, we were given a system of three equations involving derivatives of three variables:

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

The approach here included differentiating and substituting equations systematically to eventually form a single third-order differential equation. This step-by-step process is crucial, as it allows us to deeply understand the interrelationships between the different variables and the nature of their change over time. By following systematic elimination, we reduce the interdependence of complex systems, making the path to solving simpler and more direct.

Third-order Differential Equation

A third-order differential equation involves the third derivative of a variable or function. Unlike first or second-order equations, these are higher-order and naturally more complex. They describe systems or situations where the rate of change itself is changing, and not just the position or the speed of something.

In our context, after employing systematic elimination on the system of equations, we derived the third-order differential equation:\[ D^3 x = x \].

This equation means that the third derivative of \( x \) is equal to \( x \) itself. To solve such equations, one commonly uses methods that involve characteristic equations. Understanding a third-order equation is about gaining insight into how a variable behaves with respect to its own higher-level changes over time, revealing patterns and properties that are not obvious at lower orders.

Characteristic Equation Roots

Characteristic equation roots are integral to solving higher-order differential equations. After forming a third-order differential equation, these roots provide solutions that define the behavior of the system. To find these roots, we first write the characteristic equation by translating the differential equation into an algebraic form.

For the differential equation \(D^3 x - x = 0\), the characteristic equation is found by assuming solutions of the form \(x(t) = e^{rt}\), which leads to the polynomial equation, referred to as the characteristic equation.

• The roots for our example were: \(1, -\frac{1}{2} \pm i\frac{\sqrt{3}}{2}\).

Each root corresponds to a component of the solution, influencing the form and behavior of the solution functions. Real roots indicate exponential growth or decay, while complex conjugate roots describe oscillatory behavior. Recognizing and interpreting these roots allows the construction of a general solution, combining terms related to these roots to account for the full range of possible dynamics within the system.