Question

indicate how they can be employed to solve initial value problems with periodic forcing terms. Find the formal solution of the initial value problem $$ y^{\prime \prime}+\omega^{2} y=f(t), \quad y(0)=0, \quad y^{\prime}(0)=0 $$ where \(f\) is periodic with period \(2 \pi\) and $$ f(t)=\left\\{\begin{aligned} 1, & 0

Short answer

Question: Determine the formal solution of the initial value problem for the given second-order ordinary differential equation $y^{\prime\prime}+\omega^2 y = f(t)$, with initial conditions $y(0) = 0$ and $y^{\prime}(0) = 0$. The forcing function $f(t)$ is piecewise and has a Fourier representation; the forcing function is also periodic with a period of $2\pi$. Answer: The formal solution of the initial value problem is $$y(t) = \sum_{n=1}^{\infty} \frac{2}{n(1-n^2\omega^2)} \sin(n\omega t).$$

Step-by-step solution

Write the Fourier series for \(f(t)\)

Since \(f(t)\) is periodic with period \(2\pi\), we can represent it as a Fourier series: $$ f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos (n\omega t) + b_n \sin (n\omega t) \right] $$ where $$ a_0 = \frac{1}{\pi} \int_{0}^{2\pi} f(t) dt $$ $$ a_n = \frac{1}{\pi} \int_{0}^{2\pi} f(t) \cos (n\omega t) dt $$ $$ b_n = \frac{1}{\pi} \int_{0}^{2\pi} f(t) \sin (n\omega t) dt $$ Now we can compute the coefficients of the Fourier series: $$ a_0 = \frac{1}{\pi} \int_{0}^{\pi} 1 dt = 1 $$ $$ a_n = \frac{1}{\pi} \int_{0}^{\pi} \cos (n\omega t) dt = 0 \quad (n \geq 1) $$ $$ b_n = \frac{1}{\pi} \left(\int_{0}^{\pi} \sin (n\omega t) dt - \int_{\pi}^{2\pi} \sin (n\omega t) dt\right) = \frac{2}{n} \quad (n\geq 1) $$ Thus, the Fourier series for \(f(t)\) is: $$ f(t) = \frac{1}{2} + \sum_{n=1}^{\infty} \frac{2}{n}\sin (n\omega t)

Solve the homogeneous ODE

The homogeneous ODE is given by: $$ y^{\prime\prime}+\omega^2 y = 0 $$ The characteristic equation is: $$ r^2 + \omega^2 = 0 $$ with roots \(r = \pm\omega i\). Thus, the general solution of the homogeneous ODE is given by: $$ y_h(t) = c_1 \cos(\omega t) + c_2 \sin(\omega t)

Find a particular solution of the non-homogeneous ODE

We can use the method of undetermined coefficients to find a particular solution for the given non-homogeneous ODE: $$ y_p(t) = \sum_{n=1}^{\infty} \left[A_n \cos(n\omega t) + B_n \sin(n\omega t)\right] $$ Now, we can differentiate \(y_p(t)\) twice to get the second derivative: $$ y_p^{\prime\prime}(t) = \sum_{n=1}^{\infty} \left[-n^2\omega^2 A_n \cos(n\omega t) - n^2\omega^2 B_n \sin(n\omega t)\right] $$ Substituting \(y^{\prime\prime}(t)\) and \(y(t)\) into the non-homogeneous ODE, we obtain: $$ \sum_{n=1}^{\infty} \left[-n^2\omega^2 (A_n\cos(n\omega t) + B_n \sin(n\omega t))\right] + \omega^2 \left[A_n \cos(n\omega t) + B_n \sin(n\omega t)\right] = f(t) $$ Comparing the coefficients of the trigonometric functions, we have: $$ \frac{2}{n}\sin(n\omega t) = (1-n^2\omega^2)B_n\sin(n\omega)t $$ Thus, \(B_n = \frac{2}{n(1-n^2\omega^2)}\) for \(n\geq 1\) and \(A_n=0\). The particular solution is: $$ y_p(t) = \sum_{n=1}^{\infty} \frac{2}{n(1-n^2\omega^2)} \sin(n\omega t)

Write the general solution of the ODE in terms of the initial conditions

Now let's combine the general solution of the homogeneous and particular solutions: $$ y(t) = y_h(t) + y_p(t) $$ To determine the unknown coefficients \(c_1\) and \(c_2\), we must satisfy the initial conditions: $$ y(0) = c_1 = 0 \quad (1) $$ $$ y^{\prime}(0) = \omega c_2 = 0 \quad (2) $$ From equation (1), we get \(c_1 = 0\). Similarly, from equation (2), we find \(c_2 = 0\). As a result, the formal solution of the initial value problem is: $$ y(t) = \sum_{n=1}^{\infty} \frac{2}{n(1-n^2\omega^2)} \sin(n\omega t)

Key concepts

Understanding Periodic Forcing Terms

When studying the dynamics of systems, we often encounter situations where an external force is applied at regular intervals. These are known as periodic forcing terms. A classic example is a swinging pendulum subjected to regular pushes. In mathematics, such forces that repeat their values at regular intervals are described as periodic functions.

In the context of differential equations, a periodic forcing term can cause the system to oscillate in a pattern that eventually repeats itself. Considering the initial value problem in the exercise, the function f(t), a piecewise function alternating between 1 and -1, represents such a periodic force acting on the system with a period of 2π. The challenge here is to determine the impact of this periodic force on the behavior of the system and find a solution to the differential equation that takes this force into account.

Periodic forcing is crucial as it can lead to complex phenomena such as resonance, where the amplitude of oscillations becomes particularly large at certain frequencies. Understanding how to solve problems with periodic forcing terms is vital in physics and engineering, where systems often experience repetitive influences.

Employing Fourier Series in Solving Differential Equations

The Fourier series is a powerful mathematical tool that can represent any periodic function as an infinite sum of sines and cosines. The idea is that even complex, discontinuous functions can be approximated by adding up simple, smooth sine and cosine waves of different frequencies and amplitudes.

Here is how it’s applied in the exercise: Firstly, the coefficients a_n and b_n are calculated using integrals of the periodic function f(t) times sine or cosine functions. These coefficients represent the amplitude of the sine or cosine components at various frequencies. Once you have these coefficients, the function f(t) can be expressed as a Fourier series, which simplifies the original problem.

Incorporating the Fourier series into differential equations allows us to solve complex non-homogeneous ordinary differential equations (ODEs) with periodic forcing terms. The key lies in determining the Fourier coefficients, which, when substituted back into the ODE, lead to a series solution that can satisfy the initial conditions of the problem. Understanding Fourier series is not only essential in mathematics but also forms the basis for many applications in signal processing and physics.

Solving Non-Homogeneous Ordinary Differential Equations

Differential equations that include terms without the unknown function or its derivatives are called homogeneous. Conversely, equations that have additional functions or terms are non-homogeneous. The method to solve these typically involves finding a particular solution to the non-homogeneous part and then adding it to the general solution of the homogeneous equation.

In the exercise provided, the non-homogeneous term is the periodic forcing function f(t). To solve for the particular solution, the method of undetermined coefficients is applied, which in this case involves a series representation using the Fourier series components as a guide. After obtaining the particular solution, you combine it with the homogeneous solution. This combination, called the general solution, accounts for both the natural behavior of the system (the homogeneous part) and the effects of the external force (the particular or non-homogeneous part).

Lastly, the initial conditions are applied to determine any constants in the homogeneous solution, leading to the complete solution that describes the motion of the system from start to onwards, considering the influence of the periodic forcing term. Understanding how to tackle non-homogeneous ODEs is of enormous benefit for engineers and scientists dealing with real-world phenomena where external forces or sources are often present.