91Ó°ÊÓ

The characteristic equation is a fundamental concept in solving linear differential equations with constant coefficients. It arises when we look for solutions of the form \(x = e^{rt}\). This transforms the differential equation into an algebraic equation, which we can solve for \(r\). In our case, from the homogeneous equation \(\frac{d^2 x}{dt^2} + 4 \frac{dx}{dt} + 13x = 0\), substituting \(x = e^{rt}\) gives us the characteristic equation as follows:

\[r^2 + 4r + 13 = 0\] We solve this quadratic equation using the quadratic formula:

\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For \(a = 1\), \(b = 4\), and \(c = 13\), it solves to give:\[r = -2 \pm 3i\]
These roots are complex, indicating that the solutions will involve exponential functions combined with sine and cosine terms.

The homogeneous equation is a differential equation set to zero (i.e., no external forcing function). For our problem, it is:

\[\frac{d^2 x}{dt^2} + 4 \frac{dx}{dt} + 13x = 0\]

The solutions to the characteristic equation \(r = -2 \pm 3i\) give rise to the homogeneous solution:

\[x_h(t) = e^{-2t}(C_1 \cos(3t) + C_2 \sin(3t))\]

Here \(C_1\) and \(C_2\) are constants to be determined by the initial and boundary conditions. This form combines exponential decay, \(e^{-2t}\), and oscillatory components, \(\cos(3t)\) and \( \sin(3t)\). These functions represent the general behavior of the system without any external input.

The particular solution deals with finding a specific solution to the non-homogeneous differential equation. In this case, the non-homogeneous term is a piecewise function \(f(t)\). We use different methods to find this solution, like the method of undetermined coefficients or variation of parameters. Since \(f(t)\) is piecewise:

• For \(0 \leq t < \pi\): The forcing function \(f(t) = 1\). We assume a constant particular solution \(x_p(t) = \frac{1}{13}\).


• For \(\pi \leq t < 2\pi\): Here \(f(t) = 1 - t\). We assume the particular solution \(x_p(t) = \frac{1}{13} - \frac{t}{4}\).


• For \(t \geq 2\pi\): \(f(t) = 0\), which makes the particular solution \(x_p(t) = 0\) and only the homogeneous solution contributes to the result.

Combining these with boundary conditions ensures the solution is continuous and differentiable across intervals.

A piecewise function defines different expressions for different intervals of the independent variable. In our problem, the function \(f(t)\) is defined as:

\[ f(t) = \begin{cases}1, & 0 \leq t < \pi \1 - t, & \pi \leq t < 2\pi \0, & t \geq 2\pi\end{cases} \]

This means we need to solve the differential equation for each interval separately:

• For \(0 \leq t < \pi\): \(f(t) = 1\).


• For \(\pi \leq t < 2\pi\): \(f(t) = 1 - t\).


• For \(t \geq 2\pi\): \(f(t) = 0\).

The solutions for these intervals are then combined together, ensuring they fit together smoothly at \(t = \pi\) and \(t = 2\pi\). Piecewise functions make our solutions more flexible and realistic, allowing us to model complex real-world situations.