91Ó°ÊÓ

Fourier series provide a powerful tool to express complex periodic functions using simpler trigonometric functions. In essence, any periodic function can be broken down into an infinite sum of sines and cosines. This decomposition helps in analyzing the behavior of the function and solving differential equations.
In our exercise, the function \( F(t) \) is a periodic odd function defined by \( F(t) = t-t^2 \) for \( 0 < t < 1 \). Since the function is odd, its Fourier series consists only of sine terms:

• The formula: \( F(t) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi t}{L}\right) \)
• The period \( L \) is 1. Thus, sine functions account for the symmetry.
• Compute the coefficients \( b_n \) using the integral \( b_n = 2 \int_0^1 (t-t^2)\sin(n\pi t)\,dt \)

This Fourier series representation is key because it transforms the problem of solving a differential equation involving \( F(t) \) into simpler equations for each frequency component.

Differential equations are equations that involve derivatives of functions. They are crucial in modeling a wide range of real-world phenomena. In this exercise, we have the second-order linear differential equation:

• \( x'' + 10x = F(t) \)
• This equation is forced with the periodic function \( F(t) \).

The goal is to find a steady periodic solution \( x_{sp}(t) \) that satisfies this equation. By utilizing the Fourier series representation of \( F(t) \), we substitute it into the differential equation and solve for the coefficients \( c_n \) in the solution:

• Assume \( x_{sp}(t) = \sum_{n=1}^{\infty} c_n \sin(n\pi t) \)
• Solve for \( c_n \) using \( c_n = \frac{b_n}{10 - (n\pi)^2} \)

Each component can be solved independently, thanks to the linearity of the differential equation, which significantly simplifies the process.

Integration by Parts is an essential technique in calculus used to integrate products of functions. It's particularly helpful when dealing with integrals that arise in the context of Fourier series:

• Based on the formula: \( \int u \, dv = uv - \int v \, du \)
• This approach breaks down complex integrals into simpler parts.

In our exercise, we encounter the need to compute \( b_n \) using the formula:

• \( b_n = 2 \int_0^1 (t\sin(n\pi t) - t^2\sin(n\pi t)) \, dt \)
• Each term requires the application of integration by parts twice.

By strategically choosing \( u \) and \( dv \) in each term, the integration becomes manageable. Repeating this process each time ensures that we accurately calculate the coefficients necessary for the Fourier series.

A computer algebra system (CAS) is a software tool designed to perform symbolic mathematical computations. These systems can solve equations, perform algebraic manipulations, and graph functions, making them indispensable in mathematical problem-solving.

• In our exercise, a CAS is used to plot the series solution for the differential equation.
• By inputting the series representation of \( x_{sp}(t) \), we can observe the oscillation behavior visually.

When using a CAS, sum enough terms in the series to achieve a smooth and accurate visual representation of \( x_{sp}(t) \). The visual helps confirm that the solution behaves as expected, providing intuition and verification beyond the algebraic expressions.