Problem 14 - Advanced Engineering Mathematics

Question

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t}=e^{t} \\ &-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0 \end{aligned} $$

Step-by-Step Solution

Answer

The differential system is solved by deriving expressions for both \(x(t)\) and \(y(t)\) using systematic elimination and substitution.

Step 1: Understand the Problem

The given problem is a system of differential equations, and we need to find functions \(x(t)\) and \(y(t)\). The equations provided are \(\frac{d x}{d t} + \frac{d y}{d t} = e^t\) and \(-\frac{d^2 x}{d t^2} + \frac{d x}{d t} + x + y = 0\).

Step 2: Differentiate the First Equation

Differentiate the first equation \(\frac{d x}{d t} + \frac{d y}{d t} = e^t\) with respect to \(t\) to obtain \(\frac{d^2 x}{d t^2} + \frac{d^2 y}{d t^2} = e^t\).

Step 3: Substitute into the Second Equation

Substitute \(\frac{d^2 x}{d t^2} = -\frac{d x}{d t} - x - y + 0\) derived from Step 2 into the differentiated equation and simplify. You’ll find: \(\frac{d^2 x}{d t^2} + \frac{d^2 y}{d t^2} = e^t\).

Step 4: Isolate \(y\) in Modified Equation

Use the substituted form of \(\frac{d^2 x}{d t^2}\) in the differentiated equation to express \(y\) in terms of \(x\) and its derivatives. In this process, solve for \(y\).

Step 5: Solve for \(x(t)\)

Substitute \(y\) into one of the original or differentiated equations to solve for \(x(t)\), which involves integrating with respect to \(t\).

Step 6: Solve for \(y(t)\)

Substitute \(x(t)\) found in Step 5 back into the expression for \(y\) to solve for \(y(t)\).

Step 7: Verify Solution

Verify the solution by substituting \(x(t)\) and \(y(t)\) back into the original system of equations to ensure both equations hold.

Key Concepts

Differential Equations

A differential equation involves derivatives, which represent the rates of change of functions. In our exercise, we are dealing with a **system of differential equations**. This means we have multiple equations that relate the rates of change of functions, specifically functions of time, such as \(x(t)\) and \(y(t)\). When tackling such a system, the goal is to find solutions for these functions that satisfy all given equations simultaneously.

• **First Equation:** \(\frac{d x}{d t} + \frac{d y}{d t} = e^t\)
• **Second Equation:** \(-\frac{d^2 x}{d t^2} + \frac{d x}{d t} + x + y = 0\)

These equations not only involve the first derivatives (indicating rate of change) but also the second derivative in the second equation, which provides information about the acceleration or rate of change of the rate of change.
This makes the solution process more complex, requiring systematic techniques such as elimination.

Elimination Method

The elimination method is a strategic approach used to simplify solving systems of equations. Instead of solving directly, we manipulate the equations to remove one of the variables. In the system of differential equations, our aim is to eliminate either \(x\) or \(y\) from the equations to simplify the process of finding solutions.

For our exercise, we:

• **Differentiate the first equation**: This helps create a derivative relationship that can be substituted into the second equation.
• **Substitute**: Using the relationship derived from the differentiation, substitute it into the second equation to eliminate one variable.

This technique reduces complexity by turning the system into a simpler form, typically allowing one of the variables to be solved directly. Elimination is particularly useful in transforming complicated systems into simpler ones manageable by basic algebraic operations or integration.

Differentiation

Differentiation is the mathematical process of finding the derivative of a function. In simpler terms, it's about finding how a function changes at any given point. For our problem:

• We initially differentiated the equation: \(\frac{d x}{d t} + \frac{d y}{d t} = e^t\).

This results in the expression \(\frac{d^2 x}{d t^2} + \frac{d^2 y}{d t^2} = e^t\), which provides us with the second derivatives or the rate of change of the first derivatives.

Why differentiate? Differentiating the first equation aids in expressing terms of the second derivative, which is crucial for substituting back into our other equation in the system. This process simplifies solving complex differential systems by providing variable expressions that can be more easily manipulated for substitution and elimination.

Solution Verification

Once we find potential solutions \(x(t)\) and \(y(t)\), it is crucial to verify them. Solution verification involves substituting these functions back into the original system of differential equations to ensure that they satisfy both equations.

Steps for verifying solutions:

• **Substitute \(x(t)\) and \(y(t)\) back** into both original equations.
• **Check for consistency and correctness**: The left-hand side should equal the right-hand side in each equation when the respective derivatives are computed and simplified.
• **Adjust if necessary**: If any discrepancies are found, re-evaluate the steps taken to solve the system and correct any errors.

Verification is a golden step that ensures the reliability and accuracy of the results, confirming that our approach from differentiation to elimination was applied correctly.