Problem 17 - Advanced Engineering Mathematics

Question

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 solutions are \( x(t) = c_1 e^t + c_2 e^{-\frac{1}{2}t} \cos(\frac{\sqrt{3}}{2}t) + c_3 e^{-\frac{1}{2}t} \sin(\frac{\sqrt{3}}{2}t) \), with \( y(t) \) and \( z(t) \) derived from it.

Step 1: Equate the Derivatives

First, we write down the system of equations given in terms of derivatives: \( Dx = y \) \( Dy = z \) \( Dz = x \). Our goal is to systematically eliminate variables to find a single equation involving only one of the variables.

Step 2: Eliminate y

Take the derivative of the equation \( Dx = y \), which gives us \( D^2x = Dy \). Using the second equation \( Dy = z \), substitute \( D^2x = z \). Now, we have two equations: \( D^2x = z \) \( Dz = x \).

Step 3: Substitute to Find Single Variable Equation

Substitute \( z \) from \( D^2x = z \) into \( Dz = x \) to obtain \( D(D^2x) = x \). This simplifies to the third derivative equation: \( D^3x = x \).

Step 4: Solve the Single Differential Equation

The equation \( D^3x = x \) is a third-order ordinary differential equation. The characteristic equation is \( r^3 = 1 \). The roots of this equation are \( r = 1, -\frac{1}{2} + i\frac{\sqrt{3}}{2}, -\frac{1}{2} - i\frac{\sqrt{3}}{2} \).

Step 5: Write the General Solution for x

From the characteristic equation, the solution for \( x(t) \) can be written as: \( x(t) = c_1 e^{t} + c_2 e^{-\frac{1}{2}t} \cos(\frac{\sqrt{3}}{2}t) + c_3 e^{-\frac{1}{2}t} \sin(\frac{\sqrt{3}}{2}t) \), where \( c_1, c_2, \) and \( c_3 \) are constants.

Step 6: Find Solutions for y and z

Now, use the relationships \( Dx = y \) and \( Dy = z \) to find \( y(t) \) and \( z(t) \): - Differentiate \( x(t) \) to get \( y(t) \). - Differentiate \( y(t) \) to obtain \( z(t) \).

Key Concepts

Systematic Elimination

Systematic elimination is a method used to solve a system of equations by reducing the number of variables one by one until you are left with an equation involving only a single variable. This process simplifies the system, allowing for easier integration and solution.

• Start by organizing the equations so they contain variables to be eliminated sequentially.
• Take derivatives of the equations as necessary to match variables and eliminate step by step.
• Transform the system into a single differential equation.

In the exercise, we began by noting the derivatives in the given system. By taking derivatives and substituting terms like in solving for \( D^2x = z \) then replacing \( z \) in \( Dz = x \), we gradually converge on a single equation \( D^3x = x \). This systematic approach makes it manageable to find a solution to complex systems.

Ordinary Differential Equation

An ordinary differential equation (ODE) is an equation containing a function of one independent variable and its derivatives. The derivatives indicate the rate of change.
In our system, we are working with a third-order ODE: \( D^3x = x \). Here, the third derivative of \( x \) is equal to \( x \) itself.

• Ordinary Differential Equations involve only one independent variable.
• They can be first, second, or even higher order, indicating the highest derivative present in the equation.
• ODEs describe dynamic systems and can model phenomena in physics, engineering, and other fields.

For our problem, solving this third-order ODE involves translating the ODE into a characteristic equation, which will then help in finding solutions specific to the function \( x(t) \).

Characteristic Equation

The characteristic equation arises from an ordinary differential equation by assuming a trial solution of exponential form, typically \( x(t) = e^{rt} \), where \( r \) represents a constant.
In our case, from \( D^3x = x \), the trial leads to the characteristic equation \( r^3 = 1 \). To find solutions:

• Solve for the roots of the polynomial equation.
• The nature of roots (real and distinct, real and repeated, or complex) will indicate different forms of the solution.

The roots \( r = 1 \), \( -\frac{1}{2} + i\frac{\sqrt{3}}{2} \), and \( -\frac{1}{2} - i\frac{\sqrt{3}}{2} \) include both real and complex values, showing that the solution will include exponential, cosine, and sine components. This multiplicity in root types enriches the structure of the general solution.

General Solution

The general solution of a differential equation takes into account all possible solutions that satisfy the equation, using the different roots found from the characteristic equation.
For the third-order ODE \( D^3x = x \), we found roots \( 1 \), \( -\frac{1}{2} + i\frac{\sqrt{3}}{2} \), and \( -\frac{1}{2} - i\frac{\sqrt{3}}{2} \). The general solution for \( x(t) \) becomes:

• \( x(t) = c_1 e^t + c_2 e^{-\frac{1}{2}t} \cos(\frac{\sqrt{3}}{2}t) + c_3 e^{-\frac{1}{2}t} \sin(\frac{\sqrt{3}}{2}t) \)
• The constants \( c_1 \), \( c_2 \), and \( c_3 \) are determined by initial conditions or boundary values.

Lastly, using the derived formula for \( x(t) \), we differentiate to find corresponding \( y(t) = Dx \) and \( z(t) = Dy \), completing the solutions for the entire system. This comprehensive solution format reflects the integrated behavior of the original system.