Problem 21 - Advanced Engineering Mathematics

Question

In Problems 21-24, verify that the vector \(\mathbf{X}_{p}\) is a particular solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=x+4 y+2 t-7 \\ &\frac{d y}{d t}=3 x+2 y-4 t-18 ; \quad \mathbf{X}_{p}=\left(\begin{array}{r} 2 \\ -1 \end{array}\right) t+\left(\begin{array}{l} 5 \\ 1 \end{array}\right) \end{aligned} $$

Step-by-Step Solution

Answer

The vector \( \mathbf{X}_{p} \) satisfies the system, so it is a particular solution.

Step 1: Represent X_p as x(t) and y(t)

The particular solution vector \( \mathbf{X}_{p} \) is given as \( \begin{pmatrix} 2 \ -1 \end{pmatrix} t + \begin{pmatrix} 5 \ 1 \end{pmatrix} \). We can express this as two separate functions: 1. \( x(t) = 2t + 5 \)2. \( y(t) = -t + 1 \)

Step 2: Calculate derivatives of x(t) and y(t)

Find the derivative of \( x(t) \) and \( y(t) \) with respect to \( t \):- \( \frac{dx}{dt} = \frac{d}{dt}(2t + 5) = 2 \)- \( \frac{dy}{dt} = \frac{d}{dt}(-t + 1) = -1 \)

Step 3: Substitute x(t) and y(t) into original equations

Substitute \( x(t) = 2t + 5 \) and \( y(t) = -t + 1 \) into the original system of equations:\[ \frac{dx}{dt} = x + 4y + 2t - 7 \]\[ 2 = (2t + 5) + 4(-t + 1) + 2t - 7 \]

Step 4: Simplify first equation

Simplify the equation \( 2 = (2t + 5) + 4(-t + 1) + 2t - 7 \):- \( 2 = 2t + 5 - 4t + 4 + 2t - 7 \)- Combine like terms: \( 2 = 0 \)- Since both sides satisfy the equation, it holds true.

Step 5: Substitute into second equation

Substitute the same \( x(t) \) and \( y(t) \) into the second equation:\[ \frac{dy}{dt} = 3x + 2y - 4t - 18 \]\[ -1 = 3(2t + 5) + 2(-t + 1) - 4t - 18 \]

Step 6: Simplify second equation

Simplify the equation \( -1 = 3(2t + 5) + 2(-t + 1) - 4t - 18 \):- \( -1 = 6t + 15 - 2t - 2 - 4t - 18 \)- Combine like terms: \( -1 = 0 \)- Since both sides satisfy the equation, it holds true.

Key Concepts

Particular Solution Verification

When dealing with systems of differential equations, a particular solution is a solution that satisfies the given differential equations. In this context, the vector \( \mathbf{X}_{p} \) represents such a particular solution. To verify that \( \mathbf{X}_{p} \) is a legitimate solution, we check if it satisfies both differential equations in the system.

The verification process involves substituting the expressions for \( x(t) \) and \( y(t) \) derived from \( \mathbf{X}_{p} \) back into the original differential equations, and subsequently checking if the equations hold true.

• Calculate the derivatives of each component with respect to \( t \).
• Substitute these derivatives and \( x(t) \), \( y(t) \) functions into their respective differential equations.
• Simplify both equations to examine if both sides of each equation equal each other.

If both equations are satisfied, it confirms that \( \mathbf{X}_{p} \) is indeed a particular solution. This process ensures all components meet the conditions set by the system of differential equations.

Derivatives

Derivatives measure how a function changes as its input changes, which is especially important in differential equations where we describe physical systems that evolve over time. For the given problem, derivatives help us understand the rate of change of each component \( x(t) \) and \( y(t) \).

To obtain the derivatives, we apply basic derivative rules:

• For \( x(t) = 2t + 5 \), the derivative is \( \frac{dx}{dt} = 2 \).
• For \( y(t) = -t + 1 \), the derivative is \( \frac{dy}{dt} = -1 \).

These computations are straightforward because they involve simple linear functions. Understanding and calculating derivatives are foundational skills, as they allow us to complete further analysis by comparing these rates of change to given differential equations.

Linear Algebra

Linear algebra plays a crucial role in analyzing and solving systems of differential equations. It provides the tools and conceptual framework necessary to represent, manipulate, and solve these systems. In our specific example, the solution is expressed as a linear combination of vectors, which is a key operation in linear algebra.

The particular solution \( \mathbf{X}_{p} \) can be seen as a linear combination of vectors:

• The term \( \begin{pmatrix} 2 \ -1 \end{pmatrix} t \) indicates a direction vector whose influence depends on \( t \).
• The fixed vector \( \begin{pmatrix} 5 \ 1 \end{pmatrix} \) adds a constant shift.

Recognizing these structures allows us to capitalize on linear algebra principles, such as vector addition and scaling, to perform verification and solution methods efficiently. It confirms that being able to decompose functions into such linear forms is a powerful strategy.

Differential Equations Solution Methods

Solving differential equations involves finding functions that satisfy those equations, a process central to modeling and understanding dynamic systems. Various methods can be used depending on the type of differential equations encountered.

For linear differential equations like the ones presented here, common methods include:

• **Substitution**: Plugging potential solutions into the differential equation to test validity.
• **Decomposition**: Breaking down complex parts into simpler, manageable pieces through derivatives and separate equations.
• **Verification**: Ensuring correctness by solving the components separately and confirming they satisfy the original equation.

Each method allows us to systematically approach problems, ensuring that we obtain a solution that fully respects the initial conditions and the structure dictated by the equation.