Problem 26 - Advanced Engineering Mathematics

Question

Prove that the general solution of $$ \mathbf{X}^{\prime}=\left(\begin{array}{rr} -1 & -1 \\ -1 & 1 \end{array}\right) \mathbf{X}+\left(\begin{array}{l} 1 \\ 1 \end{array}\right) t^{2}+\left(\begin{array}{r} 4 \\ -6 \end{array}\right) t+\left(\begin{array}{r} -1 \\ 5 \end{array}\right) $$ on the interval \((-\infty, \infty)\) is $$ \begin{aligned} \mathbf{X}=& c_{1}\left(\begin{array}{c} 1 \\ -1-\sqrt{2} \end{array}\right) e^{\sqrt{2} t}+c_{2}\left(\begin{array}{c} 1 \\ -1+\sqrt{2} \end{array}\right) e^{-\sqrt{2} t} \\ &+\left(\begin{array}{l} 1 \\ 0 \end{array}\right) t^{2}+\left(\begin{array}{r} -2 \\ 4 \end{array}\right) t+\left(\begin{array}{l} 1 \\ 0 \end{array}\right) . \end{aligned} $$

Step-by-Step Solution

Answer

The general solution is found by combining the homogeneous and particular solutions.

Step 1: Rewrite the Differential Equation

The given differential equation is \( \mathbf{X}^{\prime} = A \mathbf{X} + \mathbf{g}(t) \) where \( A = \begin{pmatrix} -1 & -1 \ -1 & 1 \end{pmatrix} \) and \( \mathbf{g}(t) = \begin{pmatrix} 1 \ 1 \end{pmatrix} t^2 + \begin{pmatrix} 4 \ -6 \end{pmatrix} t + \begin{pmatrix} -1 \ 5 \end{pmatrix} \). The general solution contains the homogeneous and particular solutions of this system.

Step 2: Solve the Homogeneous System

The homogeneous system is \( \mathbf{X}' = A \mathbf{X} \). Find the eigenvalues of \( A \) by solving \( \text{det}(A - \lambda I) = 0 \), which gives \( \lambda = \pm \sqrt{2} \). The eigenvectors corresponding to \( \lambda = \sqrt{2} \) and \( -\sqrt{2} \) are \( \begin{pmatrix} 1 \ -1 - \sqrt{2} \end{pmatrix} \) and \( \begin{pmatrix} 1 \ -1 + \sqrt{2} \end{pmatrix} \), respectively. Thus, the solution to the homogeneous equation is \( c_1 \begin{pmatrix} 1 \ -1-\sqrt{2} \end{pmatrix} e^{\sqrt{2} t} + c_2 \begin{pmatrix} 1 \ -1+\sqrt{2} \end{pmatrix} e^{-\sqrt{2} t} \).

Step 3: Find the Particular Solution

To find a particular solution \( \mathbf{X}_p(t) \), solve \( \mathbf{X}' = A \mathbf{X} + \mathbf{g}(t) \) using the form \( \mathbf{X}_p(t) = \mathbf{a} t^2 + \mathbf{b} t + \mathbf{c} \), where \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are constant vectors. Plug into the differential equation and match coefficients to obtain \( \mathbf{a} = \begin{pmatrix} 1 \ 0 \end{pmatrix} \), \( \mathbf{b} = \begin{pmatrix} -2 \ 4 \end{pmatrix} \), and \( \mathbf{c} = \begin{pmatrix} 1 \ 0 \end{pmatrix} \). Thus, the particular solution is \( \mathbf{X}_p(t) = \begin{pmatrix} 1 \ 0 \end{pmatrix} t^2 + \begin{pmatrix} -2 \ 4 \end{pmatrix} t + \begin{pmatrix} 1 \ 0 \end{pmatrix} \).

Step 4: Combine Solutions for General Solution

Combine the homogeneous and particular solutions to form the general solution \( \mathbf{X}(t) = c_1 \begin{pmatrix} 1 \ -1-\sqrt{2} \end{pmatrix} e^{\sqrt{2} t} + c_2 \begin{pmatrix} 1 \ -1+\sqrt{2} \end{pmatrix} e^{-\sqrt{2} t} + \begin{pmatrix} 1 \ 0 \end{pmatrix} t^2 + \begin{pmatrix} -2 \ 4 \end{pmatrix} t + \begin{pmatrix} 1 \ 0 \end{pmatrix} \), which matches the given general solution.

Key Concepts

Homogeneous Solutions

In mathematics, particularly in the study of differential equations, a homogeneous solution is one which sets the non-homogeneous part of the equation to zero. For our differential equation, this involves looking at the system as if it is "self-contained," without any external forcing or inhomogeneity. The given system is represented by \(\mathbf{X}' = A \mathbf{X}\), where \(A = \begin{pmatrix} -1 & -1 \ -1 & 1 \end{pmatrix}\).
The task is to find solutions that satisfy this homogeneous part. When we solve it, we do not yet consider the additional \(\mathbf{g}(t)\) terms. This solution helps in understanding the inherent dynamics of the system. By finding the eigenvalues and eigenvectors from matrix \(A\), we construct solutions that have exponential terms based on the eigenvalues.
Finding eigenvalues allows us to characterize how solutions behave over time, forming the basis of our general solution.

Particular Solution

A particular solution is a specific solution to a differential equation that takes into account the non-homogeneous component \(\mathbf{g}(t)\). While homogeneous solutions address the natural system behavior without external factors, the particular solution accounts for added functions, such as polynomials or exponential terms, modifying the system.
In our equation, we use a combination of polynomial terms to find a suitable particular solution. Assume a form \(\mathbf{X}_p(t) = \mathbf{a} t^2 + \mathbf{b} t + \mathbf{c}\). Substituting this into the original equation allows you to determine constants \(\mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c}\) such that the equation is satisfied.

• Using comparison of coefficients helps determine these constants, leading to the particular solution that complements the problem's non-homogeneous elements.

By adding this specific solution to the homogeneous one, we derive the complete general solution of the differential system.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are key terms crucial for solving systems of linear differential equations. The eigenvalues are values \(\lambda\) found by solving the characteristic equation derived from the determinant \(\text{det}(A - \lambda I) = 0\). Here, they help specify the growth rates of solutions.
For the given matrix \(A\), solving results in eigenvalues \(\lambda = \pm \sqrt{2}\). These particular numbers are instrumental in describing how system state vectors evolve over time. The corresponding eigenvectors represent directions along which these growths or decays occur.
Eigenvectors are found by substituting each eigenvalue into the equation \((A - \lambda I)\mathbf{v} = 0\), finding non-trivial solutions. These vectors, paired with corresponding exponential terms, form the homogeneous solution, offering a deep insight into the system's intrinsic properties.

Matrix Differential Equations

Matrix differential equations are a broad and potent class of differential equations where the derivatives are described using matrix operations. These equations are paramount in fields like physics, engineering, and economics due to their ability to model systems with multiple interacting variables.
The structure \(\mathbf{X}' = A\mathbf{X} + \mathbf{g}(t)\) is an archetype of a matrix differential equation, where the derivative \(\mathbf{X}'\) is expressed as a linear transformation through matrix \(A\) in conjunction with an additional function \(\mathbf{g}(t)\).
The process of solving such equations often involves linear algebraic techniques, including finding eigenvalues and eigenvectors, to simplify the multi-variable problem into more manageable ordinary differential equations. Thus, matrix differential equations elegantly expand classic differential methods to vector spaces, enabling the modeling of complex dynamic systems.