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 $$ y^{\prime \prime}+\omega^{2} y=\sum_{i=1}^{\infty} b_{n} \sin n t, \quad y(0)=0, \quad y^{\prime}(0)=0 $$ where \(\omega>0\) is notequal to a positive integer. How is the solution altered if \(\omega=m,\) where \(m\) is a positive integer?

Short answer

Short Answer: The solution to the given initial value problem is the sum of the homogeneous and particular solutions. The homogeneous solution in this case is zero. The particular solution is given by \(y_p(t) = \sum_{i=1}^{\infty} \frac{b_n}{\omega^2 - n^2} \sin(nt)\). However, if \(\omega\) is a positive integer \(m\), we need to redefine the particular solution to avoid a singularity in the denominator. In this case, the particular solution is given by \(y_p(t) = \sum_{i=1, i \neq m}^{\infty} a_n \sin(nt) + d_m t \sin(mt)\), where \(d_m\) is a constant to be determined for the specific case when \(n=m\).

Step-by-step solution

Determine the homogeneous solution

First, consider the homogeneous equation without the forcing terms: $$ y^{\prime \prime} + \omega^2 y = 0. $$ Given that the coefficients are constants, the homogeneous solution will have the following form: $$ y_h(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t). $$ Now we can use the initial conditions to find the constants \(C_1\) and \(C_2\):

Apply the initial conditions

Using the initial conditions \(y(0) = 0\) and \(y'(0) = 0\), we get: $$ y_h(0) = C_1 \cos(0) + C_2 \sin(0) = C_1 = 0, $$ and $$ y'_h(t) = -C_1 \omega \sin(\omega t) + C_2 \omega \cos(\omega t), $$ $$ y'_h(0) = -C_1 \omega \sin(0) + C_2 \omega \cos(0) = C_2 \omega = 0, $$ which implies \(C_2 = 0\) as well. So the homogeneous solution is $$ y_h(t) = 0. $$

Find the particular solution

We can now look for a particular solution to the inhomogeneous equation. Since the forcing term is a sum of sine functions, we will assume a solution of the form $$ y_p(t) = \sum_{i=1}^{\infty} a_n \sin(nt), $$ where we need to determine the coefficients \(a_n\). Differentiate \(y_p(t)\) twice to get $$ y^{\prime\prime}_p(t) = -\sum_{i=1}^{\infty} a_n n^2 \sin(nt). $$ Substitute \(y_p(t)\) and its second derivative into the original inhomogeneous equation: $$ -\sum_{i=1}^{\infty} a_n n^2 \sin(nt) + \omega^2 \sum_{i=1}^{\infty} a_n \sin(nt) = \sum_{i=1}^{\infty} b_n \sin(nt). $$

Equate coefficients

We now equate the coefficients of the sine functions to solve for \(a_n\): $$ (-a_n n^2 + \omega^2 a_n) \sin(nt) = b_n \sin(nt). $$ $$ a_n(\omega^2 - n^2) = b_n, \text{ for } n = 1, 2, 3, \dots $$ This gives us $$ a_n = \frac{b_n}{\omega^2 - n^2}. $$

Write the complete solution

The complete solution is the sum of the homogeneous and particular solutions: $$ y(t) = y_h(t) + y_p(t) = \sum_{i=1}^{\infty} \frac{b_n}{\omega^2 - n^2} \sin(nt). $$

Analyze the change for the case \(\omega = m\)

Observe that when \(\omega = m\) where \(m\) is a positive integer, we will have a singularity in the denominator of \(a_n\) when \(n = m\). This will require a different approach to find the particular solution for that specific term. The particular solution can be redefined as follows: $$ y_p(t) = \sum_{i=1, i \neq m}^{\infty} a_n \sin(nt) + d_m t \sin(mt), $$ where \(d_m\) is a constant to be determined for the specific case when \(n=m\).

Key concepts

Differential Equations

Differential equations are mathematical equations that describe how a particular quantity changes over time or space. They involve derivatives, which represent rates of change, and are fundamental in expressing physical laws and phenomena. Differential equations can be classified into various types based on their characteristics, including ordinary differential equations (ODEs) and partial differential equations (PDEs), as well as linear and nonlinear equations.

For example, the equation presented in the exercise, \( y^{\prime \prime} + \omega^2 y = \sum_{i=1}^{\infty} b_{n} \sin n t \), is a second-order ODE because it includes the second derivative of the function \(y(t)\) and is linear since the equation provides a straight-line relationship. Solving such an equation often requires finding a function \(y(t)\) that satisfies the equation for given initial conditions and parameters.

Periodic Forcing Terms

In the study of differential equations, periodic forcing terms refer to components that forcibly oscillate or vary in a repetitive manner over a regular time interval. These terms are responsible for driving the system's behavior away from its natural unforced state. They often appear as functions like sine or cosine, representing a continuous, predictable pattern of variation.

In the given exercise, \( \sum_{i=1}^{\infty} b_{n} \sin n t \) represents an infinite series of periodic forcing terms, where each term individually oscillates with a different frequency. When these terms are applied to a system governed by a differential equation, they can induce complex oscillatory behaviors, depending on the interaction between the system's natural dynamics (characterized by the homogeneous solution) and the forcing frequencies. The provided problem highlights the importance of finding a solution that accounts for both the inherent system properties and the effects of periodic forcing.

Homogeneous Solution

A homogeneous solution, or the complementary function, arises in the context of linear differential equations when all the terms involving the function and its derivatives are set to zero, leaving no forcing term. This represents the solution of the system when it is not subject to any external influence — its natural motion.

In our exercise, the homogeneous equation is \( y^{\prime \prime} + \omega^2 y = 0 \), and its general solution is of the form \( y_h(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t) \). This is because the characteristic equation of the differential equation has solutions that lead to circular (sinusoidal and cosinusoidal) functions of time, which describe the system's undisturbed oscillations. The constants \(C_1\) and \(C_2\) are determined by applying the initial conditions, which, in the case of the exercise, result in \(C_1 = C_2 = 0\), simplifying the homogeneous solution to zero.

Particular Solution

The particular solution of a differential equation specifically addresses the non-homogeneous part, which in our context, is the forcing term. Unlike the homogeneous solution that characterizes the system's natural behavior, the particular solution captures the response of the system due to external forces.

In order to solve for the particular solution of the equation presented in the exercise, we assume a solution \( y_p(t) = \sum_{i=1}^{\infty} a_n \sin(nt) \), which matches the form of the forcing term. The coefficients \(a_n\) of this assumed solution reflect how the system responds to each sine component of the forced vibration. They are calculated step by step by differentiating the assumed solution, substituting it into the original equation, and then equating coefficients, which gives us the expression \( a_n = \frac{b_n}{\omega^2 - n^2} \) for each term in the series. This series represents how the system absorbs and resonates with each frequency component of the periodic forcing term, thus contributing to the overall solution.