Problem 41 - Advanced Engineering Mathematics

Question

In Problems \(41-44\), use systematic elimination to solve the given system. $$ \begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t}=2 x+2 y+1 \\ &\frac{d x}{d t}+2 \frac{d y}{d t}=y+3 \end{aligned} $$

Step-by-Step Solution

Answer

\( \frac{d x}{d t} = 4x + 3y - 1 \) and \( \frac{d y}{d t} = -2x - y + 2 \).

Step 1: Label the Equations

First, let's clearly label the given system of equations for easier manipulation. We have two equations:Equation (1): \( \frac{d x}{d t} + \frac{d y}{d t} = 2x + 2y + 1 \)Equation (2): \( \frac{d x}{d t} + 2\frac{d y}{d t} = y + 3 \).

Step 2: Eliminate One of the Derivatives

Subtract Equation (1) from Equation (2) to eliminate \( \frac{d x}{d t} \):\[(\frac{d x}{d t} + 2\frac{d y}{d t}) - (\frac{d x}{d t} + \frac{d y}{d t}) = (y + 3) - (2x + 2y + 1) \]Simplifying this gives:\[\frac{d y}{d t} = -2x - y + 2\]

Step 3: Solve for \(\frac{dy}{dt}\) in Terms of \(x\) and \(y\)

Now we have an expression for \( \frac{d y}{d t} \):\[ \frac{d y}{d t} = -2x - y + 2 \]This tells us how \( y \) is changing with respect to time \( t \), expressed in terms of \( x \) and \( y \) themselves.

Step 4: Substitute Back to Find \(\frac{dx}{dt}\)

To find \( \frac{d x}{d t} \), substitute \( \frac{d y}{d t} = -2x - y + 2 \) into Equation (1):\[\frac{d x}{d t} + (-2x - y + 2) = 2x + 2y + 1\]Rearrange terms:\[\frac{d x}{d t} = 4x + 3y - 1\]

Step 5: Rewrite the System with Solved Expressions

Now we can express the original system in terms of \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) as:\[\frac{d x}{d t} = 4x + 3y - 1\]\[\frac{d y}{d t} = -2x - y + 2\]This represents the solution to the system using systematic elimination.

Key Concepts

Systematic Elimination

Systematic elimination is a method used to solve systems of differential equations by eliminating variables strategically. This technique simplifies the problem, making it easier to handle complex systems. In the given exercise:

• The equations were labeled for clarity: Equation (1) and Equation (2).
• By subtracting one equation from another, the term \( \frac{d x}{d t} \) is eliminated from the system.

After elimination, what remains is easier to solve for one of the derivatives, such as \( \frac{d y}{d t} \). This reduction makes it possible to express one of the variables solely in terms of the other, simplifying further steps in solving the system. This method highlights the importance of strategic manipulation in mathematics to derive straightforward solutions.

Coupled Differential Equations

Coupled differential equations are equations where the dependent variables are intertwined through their derivatives. In this exercise:

• Both \( x(t) \) and \( y(t) \) depend on each other's rates of change.
• The system involves equations where the derivative of one variable affects the rate of another.

By recognizing the coupling, we can manipulate the equations to isolate interdependencies. Understanding this concept reveals how variables relate within the system, and managing these relationships helps in arriving at a comprehensive solution.

Linear Differential Equations

Linear differential equations involve derivatives that are linear with respect to the unknown functions and their derivatives. In this system:

• Each equation fits into a linear form, with terms involving \( x \), \( y \), and their derivatives.
• Linearity implies that solutions can be superimposed and the equations can be handled using algebraic techniques.

The linearity allows the use of simple algebraic methods and insights from linear algebra to simplify and eventually solve the equations. It’s the structured nature of linear equations that makes them more approachable, analytical, and solvable compared to non-linear ones.

Differential Systems Solutions

Solving differential systems involves finding formulas for each variable function over time. In essence:

• The goal is to capture the behavior of \( x(t) \) and \( y(t) \) as functions of time.
• By using systematic elimination and handling the linearity, the problem is reduced to a solvable set of expressions for each variable.

In the provided solution:

• We derived expressions for both \( \frac{d x}{d t} = 4x + 3y - 1 \) and \( \frac{d y}{d t} = -2x - y + 2 \).

These results give insights into how \( x \) and \( y \) change over time, allowing us to study their dynamics and interactions. Understanding these solutions provides a foundation for exploring more complex systems. The ability to break down and solve components individually makes comprehensive analysis of differential systems possible.