Problem 16 - Advanced Engineering Mathematics

Question

Use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}2 & -1 \\ 4 & 2\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\sin 2 t \\ 2 \cos 2 t\end{array}\right) e^{2 t}\)

Step-by-Step Solution

Answer

The solution is the sum of the homogeneous and particular solutions: \(\mathbf{X} = \mathbf{X}_h + \mathbf{X}_p\).

Step 1: Find the Homogeneous Solution

First, solve the homogeneous system \(\mathbf{X}^{\prime} = A \mathbf{X}\), where \(A = \left(\begin{array}{rr}2 & -1 \ 4 & 2\end{array}\right)\). Compute the eigenvalues by finding the determinant of \(A - \lambda I = \left(\begin{array}{rr}2 - \lambda & -1 \ 4 & 2 - \lambda\end{array}\right)\). The characteristic equation is \((2-\lambda)^2 + 4 = \lambda^2 - 4\lambda + 4 + 4 = \lambda^2 - 4\lambda + 8 = 0\). Solve this for \(\lambda\).

Step 2: Calculate Eigenvalues

Solve the quadratic equation \(\lambda^2 - 4\lambda + 8 = 0\) using the quadratic formula: \(\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = 8\). This simplifies to \(\lambda = \frac{4 \pm \sqrt{16 - 32}}{2} = \frac{4 \pm 4i}{2}\). The eigenvalues are \(\lambda_1 = 2 + 2i\) and \(\lambda_2 = 2 - 2i\).

Step 3: Find Eigenvectors

For \(\lambda_1 = 2 + 2i\), solve \((A - \lambda_1 I)\vec{v} = \mathbf{0}\):\[\begin{pmatrix} -2i & -1 \ 4 & -2i \end{pmatrix}\begin{pmatrix}v_1 \ v_2\end{pmatrix} = \begin{pmatrix}0 \ 0\end{pmatrix}\]. Choose \(v_2 = 1\), leading to \(v_1 = i\). Eigenvector is \(\vec{v}_1 = \begin{pmatrix}i \ 1\end{pmatrix}\).\Similarly for \(\lambda_2\): \(\vec{v}_2 = \overline{\vec{v}_1} = \begin{pmatrix}-i \ 1\end{pmatrix}\).

Step 4: Construct Homogeneous Solution

The homogeneous solution is \(\mathbf{X}_h = c_1 e^{(2+2i)t} \begin{pmatrix}i \ 1\end{pmatrix} + c_2 e^{(2-2i)t} \begin{pmatrix}-i \ 1\end{pmatrix}\). Express using Euler's formula: \(e^{it} = \cos t + i \sin t\):\[\mathbf{X}_h = e^{2t} \left[ c_1 \begin{pmatrix}- \sin 2t \ \cos 2t\end{pmatrix} + c_2 \begin{pmatrix}\sin 2t \ \cos 2t\end{pmatrix} \right]\].

Step 5: Setup Variation of Parameters

Apply variation of parameters by assuming \(\mathbf{X}_p = \mathbf{u}(t) e^{2t}\begin{pmatrix} \sin 2t \ \cos 2t \end{pmatrix}\). Find \(\mathbf{u}(t)\) by setting \(\mathbf{X} = \mathbf{X}_h + \mathbf{X}_p\) and differentiating.

Step 6: Differentiate Assumed Particular Solution

Differentiate \(\mathbf{X} = \mathbf{X}_h + \mathbf{X}_p\) and match with original given system. Solve for \(\mathbf{u}(t)\) after substituting \(\mathbf{X}_p'\) into the nonhomogeneous equation to isolate and simplify \(\mathbf{u}(t)\).

Step 7: Compute Particular Solution

Substitute back to compute the particular solution \(\mathbf{X}_p\) with matrix operations that involve integration to solve for \(u(t)\) explicitly.

Step 8: Final Solution

Combine both the homogeneous and particular solutions: \(\mathbf{X} = \mathbf{X}_h + \mathbf{X}_p\). Use initial conditions if provided to solve for constants \(c_1\) and \(c_2\).

Key Concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is crucial when solving systems of differential equations. They represent the essential components that help transform and simplify these systems.
Let’s see how they are used. Consider the equation for determining eigenvalues: \( ext{det}(A - \lambda I) = 0 \). Here, \( A \) denotes the matrix, \( I \) is the identity matrix, and \( \lambda \) represents the eigenvalues.
In our problem, solving the characteristic polynomial \( \lambda^2 - 4\lambda + 8 = 0 \) using the quadratic formula results in complex eigenvalues \( \lambda_1 = 2 + 2i \) and \( \lambda_2 = 2 - 2i \).
Eigenvectors provide the direction associated with each eigenvalue. By substituting \( \lambda \) back into the matrix equation \( (A - \lambda I)\vec{v} = 0 \), an eigenvector \( \vec{v} \) is found. In our example, for each eigenvalue, an eigenvector is computed—\( \vec{v}_1 = \begin{pmatrix} i \ 1 \end{pmatrix} \) and its complex conjugate \( \vec{v}_2 = \begin{pmatrix} -i \ 1 \end{pmatrix} \).
These eigenvectors are critical as they define the modes of the system's response.

Homogeneous and Particular Solutions

Differential equations often require solutions comprising both homogeneous and particular parts.
The homogeneous solution (\( \mathbf{X}_h \)) solves the associated homogeneous equation \( \mathbf{X}^{\prime} = A \mathbf{X} \) without any external perturbations.
To find it, we use the exponential of eigenvalues combined with corresponding eigenvectors. Expressing them through Euler's formula allows us to handle complex values gracefully.
For a homogeneous solution: \( \mathbf{X}_h = c_1 e^{(2+2i)t} \begin{pmatrix} i \ 1 \end{pmatrix} + c_2 e^{(2-2i)t} \begin{pmatrix} -i \ 1 \end{pmatrix} \).
On the other hand, the particular solution (\( \mathbf{X}_p \)) addresses nonhomogeneous aspects—terms like \( \sin(2t)e^{2t} \) and \( 2\cos(2t)e^{2t} \) in our system.
Here, we employ a method called variation of parameters, which assumes a form for \( \mathbf{X}_p \) involving yet-to-be-determined functions, \( \mathbf{u}(t) \), and differentiates the assumed form to match the original differential equation.
The comprehensive solution becomes the sum of these two: \( \mathbf{X} = \mathbf{X}_h + \mathbf{X}_p \), which encompasses system behaviors under both inherent properties and external influences.

Euler's Formula

Euler's formula, \( e^{i\theta} = \cos \theta + i \sin \theta \), is a key tool for converting complex-number exponential functions back into real-number trigonometric functions.
Often, in solving differential equations, especially with complex eigenvalues, applying Euler's formula helps express solutions in terms of sine and cosine.
In our exercise, the homogeneous solution for \( \mathbf{X}_h \) is expressed using Euler's formula to rewrite complex exponential expressions:
\[ e^{(2+2i)t} = e^{2t}(\cos(2t) + i\sin(2t)), \]\[ e^{(2-2i)t} = e^{2t}(\cos(2t) - i\sin(2t)). \]This approach helps simplify expressions involving complex exponentials and aligns them with physical interpretations, as many systems behave like oscillations (sinusoidal components) over time.
Thus, Euler's formula is not just a mathematical curiosity but a fundamental part of framing and understanding how oscillatory systems behave in dynamic contexts.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They express the rate of change of a quantity and are fundamental to understanding dynamic systems.
In our context, differential equations model how system states change over time by linking functions of \( t \) with their derivatives. The task often involves systems of first-order linear differential equations, like \( \mathbf{X}^{\prime} = A \mathbf{X} + \mathbf{f}(t) \), where \( A \) is a matrix and \( \mathbf{f}(t) \) is a vector expressing external contributions.
The solution to this system typically includes a combination of homogeneous and particular solutions, using techniques like eigenvalue analysis and variation of parameters to solve. Understanding these allows analysts to predict system behaviors under various conditions.
Differential equations are hugely relevant across fields—from physics to finance—and mastering them is key to mathematically describing real-world changes, such as vibrations in mechanical systems or fluctuations in economic factors.