Problem 11 - Differential Equations and Linear Algebra

Question

Consider the nonhomogeneous system $$\begin{array}{l} x_{1}^{\prime}=2 x_{1}-3 x_{2}+34 \sin t \\ x_{2}^{\prime}=-4 x_{1}-2 x_{2}+17 \cos t \end{array}$$ Find the general solution to this system by first solving the associated homogeneous system, and then using the method of undetermined coefficients to obtain a particular solution. [Hint: The form of the nonhomogeneous term suggests a trial solution of the form $$\mathbf{x}_{p}(t)=\left[\begin{array}{l} A_{1} \cos t+B_{1} \sin t \\ A_{2} \cos t+B_{2} \sin t \end{array}\right]$$ where the constants \(A_{1}, A_{2}, B_{1},\) and \(B_{2}\) can be determined by substituting into the given system.]

Step-by-Step Solution

Answer

The general solution of the given nonhomogeneous system is: \(\mathbf{x}(t) = \begin{bmatrix} 3k_1 e^{-2t} - 12\cos t \\ k_1 e^{-2t} - 5\cos t + 8\sin t \end{bmatrix}\), where \(k_1\) is an arbitrary constant.

Step 1: Solve the associated homogeneous system

We are given the nonhomogeneous system: \(\begin{cases} x_{1}^{\prime} = 2x_{1} - 3x_{2} + 34\sin t \\ x_{2}^{\prime} = -4x_{1} - 2x_{2} + 17\cos t \end{cases}\) The associated homogeneous system is: \(\begin{cases} x_{1H}^{\prime} = 2x_{1H} - 3x_{2H} \\ x_{2H}^{\prime} = -4x_{1H} - 2x_{2H} \end{cases}\) Let's rewrite this homogeneous system in matrix form: \(\mathbf{x}_H'(t) = A\mathbf{x}_H(t)\), where \(A\) is a constant matrix, and \(\mathbf{x}_H'(t) = \begin{bmatrix} x_{1H}'\\x_{2H}' \end{bmatrix}\) and \(\mathbf{x}_H(t) = \begin{bmatrix} x_{1H}\\x_{2H} \end{bmatrix}\). Now, \(A = \begin{bmatrix} 2 & -3 \\ -4 & -2 \end{bmatrix}\). Next, we find the eigenvalues and eigenvectors of the matrix \(A\). The characteristic equation is: \(\text{det}(A - \lambda I) = (2 - \lambda)(-2 - \lambda) - (-3)(-4) = 0\) \((2 - \lambda)(-2 - \lambda) + 12 = \lambda^2 + 4\lambda + 4 = 0\) Let \(\lambda_1, \lambda_2\) be the eigenvalues and \(\mathbf{v}_1, \mathbf{v}_2\) be the corresponding eigenvectors. The eigenvalues are \(\lambda_1 = \lambda_2 = -2\), and the corresponding eigenvector is \(\mathbf{v}_1 = \begin{bmatrix} 3 \\1 \end{bmatrix}\). Thus, the complementary solution (general solution of the homogeneous system) is: \(\mathbf{x}_H(t) = k_1 \mathbf{v}_1 e^{\lambda_1 t} = k_1 \begin{bmatrix} 3e^{-2t} \\e^{-2t} \end{bmatrix}\), where \(k_1\) is a constant.

Step 2: Make a trial solution for the particular solution

As suggested in the hint, we choose the trial solution \(\mathbf{x}_P(t)\) as: \(\mathbf{x}_P(t) = \begin{bmatrix} A_1 \cos t + B_1 \sin t \\ A_2 \cos t + B_2 \sin t \end{bmatrix}\) To find the derivatives, we differentiate each component of the trial solution: \(\mathbf{x}_P'(t) = \begin{bmatrix} -A_1 \sin t + B_1 \cos t \\ -A_2 \sin t + B_2 \cos t \end{bmatrix}\)

Step 3: Substitute the trial solution into the given system

Now, substitute \(\mathbf{x}_P(t)\) and \(\mathbf{x}_P'(t)\) into the given nonhomogeneous system: \(-A_1 \sin t + B_1 \cos t = 2(A_1 \cos t + B_1 \sin t) - 3(A_2 \cos t + B_2 \sin t) + 34\sin t\) \(-A_2 \sin t + B_2 \cos t = -4(A_1 \cos t + B_1 \sin t) - 2(A_2 \cos t + B_2 \sin t) + 17\cos t\) Now, we equate the coefficients of the trigonometric terms: From the \(\sin t\) terms, we get: \(-A_1 + 2B_1 - 3B_2 = 34\) \(-A_2 - 4B_1 - 2B_2 = 0\) From the \(\cos t\) terms, we get: \(B_1 + 2A_1 - 3A_2 = 0\) \(B_2 - 4A_1 - 2A_2 = 17\) Solving this linear system of equations, we find the undetermined coefficients: \(A_1 = -12, A_2 = -5, B_1 = 0, B_2 = 8\). Thus, the particular solution is: \(\mathbf{x}_P(t) = \begin{bmatrix} -12 \cos t + 0 \sin t \\ -5 \cos t + 8 \sin t \end{bmatrix} = \begin{bmatrix} -12 \cos t \\ -5\cos t + 8\sin t \end{bmatrix}\)

Step 4: Combine the complementary and particular solution

Finally, we form the general solution by combining the complementary and the particular solution: \(\mathbf{x}(t) = \mathbf{x}_H(t) + \mathbf{x}_P(t) = k_1 \begin{bmatrix} 3e^{-2t} \\e^{-2t} \end{bmatrix} + \begin{bmatrix} -12\cos t\\ -5\cos t + 8\sin t\end{bmatrix}\) Thus, the general solution of the given nonhomogeneous system is: \(\mathbf{x}(t) = \begin{bmatrix} 3k_1 e^{-2t} - 12\cos t \\ k_1 e^{-2t} - 5\cos t + 8\sin t \end{bmatrix}\), where \(k_1\) is an arbitrary constant.

Key Concepts

Nonhomogeneous Systems

In a nonhomogeneous system, differential equations include terms that do not depend on the variable being solved. These terms, known as the nonhomogeneous part, can be functions of the independent variable, like \(\sin t\) or \(\cos t\), rather than zero as in homogeneous systems. Nonhomogeneous systems appear widely in science and engineering, where external forces or inputs influence the system's behavior. To find the solution of such systems, a combination of both the homogeneous and particular solutions is used, allowing the system to accommodate external influences while also reflecting its intrinsic properties.

Method of Undetermined Coefficients

The method of undetermined coefficients is a technique to find particular solutions of nonhomogeneous differential equations, especially when the nonhomogeneous part is a simple function like a polynomial, exponential, sine, or cosine. The method involves guessing a form for the particular solution with unknown coefficients. These coefficients are determined by substituting the guess into the differential equation and ensuring the equality holds. This method is preferred due to its simplicity and effectiveness in dealing with differential equations where the nonhomogeneous term is well-structured.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra and play a crucial role in solving systems of differential equations. For a matrix \(A\), an eigenvalue \(\lambda\) is a scalar satisfying the characteristic equation \(|A - \lambda I| = 0\), where \(I\) is the identity matrix. Correspondingly, an eigenvector \(\mathbf{v}\) is a non-zero vector that satisfies \(A\mathbf{v} = \lambda\mathbf{v}\). In systems of differential equations, eigenvalues determine the type of solutions (such as exponential growth or decay), while eigenvectors provide the direction of these solutions in multi-dimensional space.

General Solution of Differential Equations

The general solution of a differential equation combines both the complementary and particular solutions. For nonhomogeneous systems, it provides a complete set of solutions accommodating both the system's inherent dynamics and external influences. Given a system, the complementary solution addresses the homogeneous part, describing the natural behavior of the system without external forces. The particular solution specifically addresses the nonhomogeneous aspect. By summing these solutions, the general solution effectively models the entire system, highlighting both internal processes and external interactions.