Problem 25 - Advanced Engineering Mathematics

Question

Use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & -1 \\ 1 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{l}\cos t \\ \sin t\end{array}\right) e^{t}\)

Step-by-Step Solution

Answer

The solution is a combination of the homogeneous solution with a particular solution obtained via variation of parameters.

Step 1: Identify the Homogeneous System

We start by considering the homogeneous part of the differential equation: \( \mathbf{X}' = \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix} \mathbf{X} \). This is the associated homogeneous system of differential equations.

Step 2: Find the Eigenvalues and Eigenvectors

To find the eigenvalues, solve \( \text{det}(A - \lambda I) = 0 \) where \( A = \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix} \) and \( I \) is the identity matrix. Calculate \( \text{det}(\begin{pmatrix} 1-\lambda & -1 \ 1 & 1-\lambda \end{pmatrix}) = (1-\lambda)^2 + 1 = 0 \). The solutions \( \lambda = 1 + i\) and \( \lambda = 1 - i \) give the eigenvalues. Using these, find the corresponding eigenvectors to determine the general solution of the homogeneous system.

Step 3: Solve the Homogeneous System

Using the eigenvalues \(\lambda = 1+i\) and \(\lambda = 1-i\), compute the eigenvectors. For \(\lambda = 1+i\), substitute back into \((A - \lambda I)\mathbf{v}=0\) to find the eigenvector \(\mathbf{v}_1\). Similarly, compute for \(\lambda = 1-i\) to get \(\mathbf{v}_2\). The general solution for the homogeneous system is \( \mathbf{X}_h = c_1 e^{t}(\cos t \mathbf{v}_1 - \sin t \mathbf{v}_2)+ c_2 e^{t}(\sin t \mathbf{v}_1 + \cos t \mathbf{v}_2) \).

Step 4: Apply Variation of Parameters

The particular solution is given by \( \mathbf{X}_p = \mathbf{X}_h \int \mathbf{X}_h^{-1} \mathbf{g}(t) \, dt \), where \( \mathbf{g}(t) = e^t \begin{pmatrix} \cos t \ \sin t \end{pmatrix} \). Calculate the inverse of the fundamental matrix formed by solutions of the homogeneous system, then integrate the product \( \mathbf{X}_h^{-1} \mathbf{g}(t) \) to find \( \mathbf{X}_p \).

Step 5: Write the General Solution

The general solution is \( \mathbf{X}(t) = \mathbf{X}_h(t) + \mathbf{X}_p(t) \). Combine the homogeneous and particular solutions obtained to express \( \mathbf{X}(t) \) completely.

Key Concepts

Linear Differential Equations

Linear differential equations form the backbone of many systems in mathematics and engineering. A linear differential equation involves an unknown function and its derivatives. For systems, these equations are often written in matrix form, such as \( \mathbf{X}^{\prime} = A\mathbf{X} \), where \( A \) is a matrix and \( \mathbf{X} \) is a vector.
These equations can describe a wide range of physical systems and phenomena. The solutions to these equations provide insight into the behavior of dynamic systems.

• Matrix Form: It allows for the compact representation and manipulation of systems of equations.
• Constant Coefficients: In our exercise, the matrix \( A \) has constant entries.
• Time Variation: The non-homogeneous part varies with time and adds complexity to the solution process.

Solving these systems often entails finding solutions to both homogeneous and non-homogeneous parts.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors play a crucial role in solving linear differential equations. For a matrix \( A \), eigenvalues \( \lambda \) are scalars that provide insights into the dynamics of the system.
To find these, we solve the characteristic equation \( \det(A - \lambda I) = 0 \). This can yield real or complex values, and these help form the basis for solution methods.

• Complex Eigenvalues: In the given problem, the eigenvalues are complex (\( 1 + i \) and \( 1 - i \)), indicating oscillatory behavior.
• Eigenvectors: These vectors indicate the direction of the stretching induced by the matrix.

Using the eigenvalues, we can derive the eigenvectors by solving \((A - \lambda I)\mathbf{v} = 0\). These elements are essential for constructing the solution to the homogeneous system of differential equations.

Homogeneous Systems

A homogeneous system is one where there is no external input, simply described as \( \mathbf{X}^{\prime} = A\mathbf{X} \). Here, no forcing functions are involved, which simplifies the analysis.
The solution strategy involves using the derived eigenvalues and eigenvectors to form the general solution. For instance, with eigenvalues like \( 1 + i \), the solution incorporates exponential growth with oscillations.

• General Solution: The solution typically includes terms with \( e^{\lambda t} \), reflecting the system's natural response.
• Superposition Principle: The total solution is a combination of modes corresponding to each eigenvalue.

In the given problem, the general solution to the homogeneous system is expressed in terms of trigonometric and exponential functions, reflecting complex eigenvalues.

Particular Solution Calculation

Calculating a particular solution using variation of parameters involves including the non-homogeneous part. Considered a powerful tool, this method expands the solution to account for external influences.
Variation of parameters leverages the homogeneous solution as a baseline and adjusts it based on the non-homogeneous term. The process involves more steps, including inversion and integration.

• Fundamental Matrix: This is composed of homogeneous solutions.
• Integration: The core task is integrating the modified system, which involves matrix inverse and non-homogeneous parts.

Completing these steps provides insights into how external forces change the behavior of the system. The general solution is obtained by adding this particular solution to the homogeneous one.