Problem 27 - Advanced Engineering Mathematics

Question

Use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right) \mathbf{X}+\left(\begin{array}{c}0 \\ \sec t \tan t\end{array}\right)\)

Step-by-Step Solution

Answer

The general solution is \( \mathbf{X}(t) = c_1 \begin{pmatrix} \cos t \\ -\sin t \end{pmatrix} + c_2 \begin{pmatrix} \sin t \\ \cos t \end{pmatrix} + \begin{pmatrix} \sin t \ln |\sec t + \tan t| \\ \cos t \ln |\sec t + \tan t| \end{pmatrix} \).

Step 1: Solve the Homogeneous System

First, find the general solution to the homogeneous system \( \mathbf{X}' = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \mathbf{X} \). The matrix has eigenvalues \( \lambda = \pm i \). The corresponding eigenvectors \( \mathbf{v}_1 = \begin{pmatrix} 1 \ i \end{pmatrix} \), and \( \mathbf{v}_2 = \begin{pmatrix} 1 \ -i \end{pmatrix} \) gives a general solution: \[ \mathbf{X}_h(t) = c_1 \begin{pmatrix} \cos t \ -\sin t \end{pmatrix} + c_2 \begin{pmatrix} \sin t \ \cos t \end{pmatrix} \].

Step 2: Setup Variation of Parameters

Next, we use variation of parameters to find a particular solution to the non-homogeneous system. Assume a particular solution of the form \[ \mathbf{X}_p(t) = \mathbf{U}(t) \begin{pmatrix} \cos t & \sin t \ -\sin t & \cos t \end{pmatrix} \]. The objective is to find the vector function \( \mathbf{U}(t) = \begin{pmatrix} u_1(t) \ u_2(t) \end{pmatrix} \).

Step 3: Compute Derivatives and Substitute

Calculate the derivative \( \mathbf{X}_p^{\prime}(t) \) and substitute it along with \( \mathbf{X}_p(t) \) into \( \mathbf{X}^{\prime} = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \mathbf{X} + \begin{pmatrix} 0 \ \sec t \tan t \end{pmatrix} \). You get \[ \begin{pmatrix} u_1^{\prime} \cos t - u_2^{\prime} \sin t \ u_1^{\prime} \sin t + u_2^{\prime} \cos t \end{pmatrix} = \begin{pmatrix} 0 \ \sec t \tan t \end{pmatrix} \].

Step 4: Solve for Vector Function \( \mathbf{U}(t) \)

Solve the resulting system of equations to find \( u_1^{\prime}(t) = 0 \) and \( u_2^{\prime}(t) = \sec t \tan t \). Thus, \( u_1(t) = C \) (a constant, which can be set to zero for simplicity in finding particular solution), and \( u_2(t) = \ln |\sec t + \tan t| + D \), where \( D \) is an integration constant.

Step 5: Formulate the Particular Solution

Using the results for \( \mathbf{U}(t) \), formulate the particular solution: \[ \mathbf{X}_p(t) = \begin{pmatrix} 0 \ \ln |\sec t + \tan t| \end{pmatrix} \begin{pmatrix} \cos t & \sin t \ -\sin t & \cos t \end{pmatrix} = \begin{pmatrix} \sin t \ln |\sec t + \tan t| \ \cos t \ln |\sec t + \tan t| \end{pmatrix} \].

Step 6: Construct the General Solution

Combine the homogeneous and particular solutions to form the general solution: \[ \mathbf{X}(t) = c_1 \begin{pmatrix} \cos t \ -\sin t \end{pmatrix} + c_2 \begin{pmatrix} \sin t \ \cos t \end{pmatrix} + \begin{pmatrix} \sin t \ln |\sec t + \tan t| \ \cos t \ln |\sec t + \tan t| \end{pmatrix} \].

Step 7: Conclusion

The final solution to the original system is the sum of the homogeneous and particular solutions. Each solution component reflects both the influence of the system's natural behavior and the external input.

Key Concepts

System of Differential Equations

A system of differential equations consists of several differential equations that describe a set of unknown functions and their derivatives. In this context, we are aiming to find a way to predict how multiple variables change, given certain initial conditions.

To solve our given system, we're dealing with a matrix form: \[ \mathbf{X}' = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \mathbf{X} + \begin{pmatrix} 0 \ \sec t \tan t \end{pmatrix} \]Here, \(\mathbf{X}\) is a vector of unknown functions that we're trying to solve for, \(\mathbf{X}'\) is the derivative of these functions with respect to time \(t\), and the matrix is how these functions are interrelated. Solving such a system typically involves finding both the complementary (homogeneous) solution and a particular solution that accounts for non-homogeneous components.

Working through the system requires understanding the dynamics captured by matrix terms and the roles of external inputs, expressed here as \(\sec t \tan t\). This approach allows us to predict the behavior of complex systems over time.

Homogeneous Solution

The homogeneous solution of a differential equation system refers to solving the system without any external or non-homogeneous terms. It describes the system's natural behavior. For our system, the homogeneous part is derived by solving: \[ \mathbf{X}' = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \mathbf{X} \]To find this, we rely on the characteristic equation derived from the matrix above, which then helps us determine eigenvalues and eigenvectors.

In this case, our system matrix leads to complex eigenvalues \(\lambda = \pm i\). These eigenvalues indicate oscillatory behavior, and they are associated with eigenvectors \(\mathbf{v}_1 = \begin{pmatrix} 1 \ i \end{pmatrix}\) and \(\mathbf{v}_2 = \begin{pmatrix} 1 \ -i \end{pmatrix}\). Using these, we can construct the general homogeneous solution: \[ \mathbf{X}_h(t) = c_1 \begin{pmatrix} \cos t \ -\sin t \end{pmatrix} + c_2 \begin{pmatrix} \sin t \ \cos t \end{pmatrix} \]This solution captures the intrinsic oscillation patterns of the system absent any forcing functions.

Particular Solution

The particular solution deals with the non-homogeneous portion of the differential equation system. Here, it means finding a solution that specifically accounts for external inputs influencing the system. This is accomplished through a technique called variation of parameters.

We assume a specific form for the particular solution: \[ \mathbf{X}_p(t) = \mathbf{U}(t) \begin{pmatrix} \cos t & \sin t \ -\sin t & \cos t \end{pmatrix} \]where \(\mathbf{U}(t) = \begin{pmatrix} u_1(t) \ u_2(t) \end{pmatrix}\) needs to be determined.

By deriving \(\mathbf{X}_p^{\prime}(t)\) and substituting back into the original differential equation, we equate terms to separate equations that give: \[ \begin{pmatrix} u_1^{\prime} \cos t - u_2^{\prime} \sin t \ u_1^{\prime} \sin t + u_2^{\prime} \cos t \end{pmatrix} = \begin{pmatrix} 0 \ \sec t \tan t \end{pmatrix} \]Solving for \(u_1^{\prime}(t)\) and \(u_2^{\prime}(t)\) leads us to the particular \(u_2(t)\) involving integration: \[ u_2(t) = \ln |\sec t + \tan t| + D \].

This quantifies how external factors cause deviation from the natural state defined by the homogeneous solution.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental to solving systems of linear differential equations. They provide insight into the system's behavior such as stability and oscillation. For differential equations, especially those with constant coefficients in systems, they define how solutions evolve over time.

Let's break down how we use them here. Given \[ \mathbf{X}' = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \mathbf{X}, \] the eigenvalue equation is \[ \text{det}(\begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}) = 0 \], which leads to eigenvalues \(\lambda = \pm i\).

These complex eigenvalues suggest solutions composed of sinusoidal functions. Each eigenvalue corresponds to a unique eigenvector, which is part of the equation's solution structure. For example, eigenvectors \(\mathbf{v}_1 = \begin{pmatrix} 1 \ i \end{pmatrix}\) and \(\mathbf{v}_2 = \begin{pmatrix} 1 \ -i \end{pmatrix}\) guide the form of the homogeneous solution.

Understanding the role of eigenvalues and eigenvectors is crucial when analyzing how solutions will behave and affect a dynamic system.