Problem 19 - A First Course in Differential Equations with Modeling Applications

Question

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}=6 y\\\ &\frac{d y}{d t}=x+z\\\ &\frac{d z}{d t}=x+y \end{aligned}$$

Step-by-Step Solution

Answer

Solve by substituting and differentiating to reduce variables, expressing everything in terms of \( x \) and \( y \).

Step 1: Differentiate the First Equation

Start by differentiating the first equation \( \frac{d x}{d t} = 6y \) with respect to time \( t \). This gives \( \frac{d^2 x}{d t^2} = 6 \frac{d y}{d t} \).

Step 2: Substitute the Second Equation

Substitute \( \frac{d y}{d t} = x + z \) from the second equation into the expression from Step 1: \( \frac{d^2 x}{d t^2} = 6(x + z) \).

Step 3: Differentiate the Second Equation

Differentiate \( \frac{d y}{d t} = x + z \) with respect to \( t \), resulting in \( \frac{d^2 y}{d t^2} = \frac{d x}{d t} + \frac{d z}{d t} \).

Step 4: Use the Third Equation

Substitute \( \frac{d z}{d t} = x + y \) into the differentiated expression from Step 3: \( \frac{d^2 y}{d t^2} = \frac{d x}{d t} + (x + y) \).

Step 5: Express in Terms of Known Quantities

Since \( \frac{d x}{d t} = 6y \), replace \( \frac{d x}{d t} \) in both \( \frac{d^2 x}{d t^2} = 6(x + z) \) and \( \frac{d^2 y}{d t^2} = \frac{d x}{d t} + (x + y) \). This yields: \( \frac{d^2 x}{d t^2} = 6x + 6z \) and \( \frac{d^2 y}{d t^2} = 6y + x + y \).

Step 6: Eliminate \( z \)

Express \( z \) from \( \frac{d^2 x}{d t^2} = 6x + 6z \) as \( z = \frac{1}{6}(\frac{d^2 x}{d t^2} - 6x) \).

Step 7: Final System of Equations

Substitute for \( z \) in the system to express all equations in terms of \( x \) and \( y \) only. This should reduce the system, allowing it to be solved systematically.

Key Concepts

System of Differential Equations

A system of differential equations is a collection of two or more differential equations involving multiple functions that are interrelated with their derivatives. Each dependent variable represents a different function, and these functions can be related to one another through their rates of change.

In the given problem, we have three equations:

• \( \frac{d x}{d t} = 6y \)
• \( \frac{d y}{d t} = x + z \)
• \( \frac{d z}{d t} = x + y \)

These equations together form a system of differential equations. Each equation specifies how one variable changes with respect to time \( t \). The aim is to determine the functions \( x(t) \), \( y(t) \), and \( z(t) \) that satisfy all three equations simultaneously.

Solving these systems often involves finding relationships between the derivatives and eliminating one or more variables. This can simplify the system to a point where standard techniques for solving differential equations, such as integration, can be applied.

Systematic Elimination Method

The systematic elimination method is a strategic approach to solving systems of differential equations. Using this method involves sequentially eliminating variables to reduce the complexity of the system.

In the provided solution, we start by differentiating equations to create links between them.

• Step 1: Differentiate the first equation \( \frac{d x}{d t} = 6y \) to get \( \frac{d^2 x}{d t^2} = 6\frac{d y}{d t} \).
• Step 2: Substitute the second equation into this result.
• Continue this process with each equation; differentiate, substitute other expressions, and simplify as needed.

By doing this, you'll gradually express all derivatives in terms of fewer variables. This simplification is key because it allows you to solve a reduced, and often a simpler system of equations. In this context, once \( z \) is eliminated from the equations, you're left with equations in terms of \( x \) and \( y \) alone, which are easier to solve.

Second Order Derivatives

Second order derivatives are the derivatives of the first derivatives, depicting the rate of change of the rate of change. In many physical systems, they represent acceleration or the curvature of a function.

In the context of the system of differential equations:

• \( \frac{d^2 x}{d t^2} \) describes how the rate of change of \( x \) with respect to time itself changes over time.
• Similarly, \( \frac{d^2 y}{d t^2} \) gives us the rate at which \( y \)'s rate of change is changing.

Differentiating each equation, as done in steps 1, 3, and so on of the solution, allows us to relate higher-order derivatives between the variables.

For instance, when you derive \( \frac{d^2 x}{d t^2} = 6(x + z) \), it links the second derivative of \( x \) directly with \( x \) and \( z \), offering insights into how these variables influence one another at an acceleration level, not just rate of change. Understanding these connections can be critical for interpreting the dynamics of the entire system.