Question

indicate how they can be employed to solve initial value problems with periodic forcing terms. Find the solution of the initial value problem $$ y^{\prime \prime}+\omega^{2} y=\sin n t, \quad y(0)=0, \quad y^{\prime}(0)=0 $$ where \(n\) is a positive integer and \(\omega^{2} \neq n^{2}\). What happens if \(\omega^{2}=n^{2} ?\)

Short answer

In summary, the solution of the initial value problem for the given second-order linear differential equation with a periodic forcing term is: $$ y(t) = \begin{cases} \frac{1}{\omega^2 - n^2}\sin(nt) & \text{if}\; \omega^2 \neq n^2 \\ \frac{t}{2n}\cos(nt) - \frac{1}{2n^2}\sin(nt) & \text{if}\; \omega^2 = n^2 \end{cases} $$ The solution varies depending on whether \(\omega^2\) is equal to \(n^2\) or not. In the case when \(\omega^2 \neq n^2\), we were able to find a particular and homogeneous solution and apply the initial conditions. However, when \(\omega^2 = n^2\), we had to use a different approach to find the particular solution and apply the initial conditions to obtain the final solution.

Step-by-step solution

Find a particular solution for the non-homogeneous equation

Since the forcing term is of the form \(\sin(nt)\), we assume there is a particular solution in the form of \(y_p(t)=A\sin(nt)+B\cos(nt)\). Differentiating \(y_p(t)\) with respect to \(t\), we have: $$ y_p'(t)=nA\cos(nt)-nB\sin(nt) $$ Differentiating once more, we obtain: $$ y_p''(t)=-n^{2}A\sin(nt)-n^{2}B\cos(nt) $$ Plug \(y_p(t)\) and its derivatives into the given differential equation: $$ (-n^{2}A\sin(nt)-n^{2}B\cos(nt))+\omega^{2}(A\sin(nt)+B\cos(nt))=\sin(nt) $$ This equation must hold for all values of \(t\). Therefore, we can equate the coefficients of the sine and cosine terms: $$ (-n^2 + \omega^2)A = 1 \Rightarrow A = \frac{1}{\omega^2 - n^2} $$ $$ (-n^2 + \omega^2)B = 0 \Rightarrow B = 0 $$ Thus, the particular solution is: $$ y_p(t) = \frac{1}{\omega^2 - n^2}\sin(nt) $$

Find the general solution of the homogeneous equation

The homogeneous equation is: $$ y''+\omega^2y=0 $$ This is a second-order linear homogeneous equation with constant coefficients, and its characteristic equation is: $$ r^2+\omega^2=0 $$ The roots of this equation are \(r=\pm\omega i\). According to the theory of second-order linear differential equations, the general solution of the homogeneous equation is given by: $$ y_h(t) = C_1\sin(\omega t)+C_2\cos(\omega t) $$

Use the initial conditions to determine the constants

The general solution of the non-homogeneous equation is the sum of the particular and homogeneous solutions: $$ y(t) = \frac{1}{\omega^2 - n^2}\sin(nt) + C_1\sin(\omega t) + C_2\cos(\omega t) $$ Apply the initial conditions \(y(0) = 0\) and \(y'(0) = 0\), we have: $$ 0 = \frac{1}{\omega^2 - n^2}\sin(0) + C_1\sin(0) + C_2\cos(0) \Rightarrow C_2 = 0 $$ and $$ 0 = n\frac{1}{\omega^2 - n^2}\cos(0) + \omega C_1\cos(0) - \omega C_2\sin(0) \Rightarrow C_1 = 0 $$ Thus, the solution of the initial value problem is: $$ y(t) = \frac{1}{\omega^2 - n^2}\sin(nt) $$

Discuss the case when \(\omega^2 = n^2\)

If \(\omega^2 = n^2\), the particular solution we found in Step 1 becomes undefined due to division by zero. In this case, the differential equation becomes a second-order linear homogeneous equation with constant coefficients: $$ y''+n^2y=\sin(nt) $$ Using a different approach, such as variation of parameters, we could find that the particular solution in this case is: $$ y_p(t) = \frac{t}{2n}\cos(nt) - \frac{1}{2n^2}\sin(nt) $$ The general solution would then be the sum of this particular solution and the homogeneous solution: $$ y(t) = \frac{t}{2n}\cos(nt) - \frac{1}{2n^2}\sin(nt) + C_1\sin(nt) + C_2\cos(nt) $$ Applying the initial conditions in this case, we would find that \(C_1 = 0\) and \(C_2 = 0\), so the solution for the initial value problem when \(\omega^2 = n^2\) is: $$ y(t) = \frac{t}{2n}\cos(nt) - \frac{1}{2n^2}\sin(nt) $$

Key concepts

Periodic Forcing Terms

When dealing with differential equations, particularly in the context of physical systems, one often encounters periodic forcing terms. These are inputs or external forces that vary with time in a periodic fashion, such as \( \sin(nt) \) or \( \cos(nt) \) where \( n \) is a positive integer representing the frequency of oscillation. The effects of these terms on the system's behavior are crucial as they can lead to resonances when the frequency of the forcing term matches a natural frequency of the system.

In the given exercise, we see that the differential equation has a periodic forcing term, \( \sin(nt) \) with an initial value problem setting. This introduces complexities in finding the solution but follows a structured approach involving both homogeneous and particular solutions.

Second-order Linear Differential Equations

A second-order linear differential equation is an equation that involves the second derivative of the unknown function and is of the form \( y'' + p(t)y' + q(t)y = g(t) \), where \( p(t) \) and \( q(t) \) are functions of \( t \) and \( g(t) \) represents the non-homogeneous part if present.

These equations are central in many areas of applied mathematics and physics, such as classical mechanics and electrical engineering. The solution generally involves two stages: finding the homogeneous solution (\text usually associated with the system's natural behavior) and then finding a particular solution (\text that accounts for the presence of external forces or periodic inputs). In this exercise, the second-order linear differential equation with periodic forcing terms requires such a treatment, and we solve it using a methodical approach.

Characteristic Equation

The characteristic equation is a vital element in solving homogeneous second-order linear differential equations. It's derived by substituting a trial solution of the form \( e^{rt} \) into the homogeneous part of the differential equation, resulting in an algebraic equation in terms of \( r \), known as the characteristic equation, which in this case is \( r^2 + \omega^2 = 0 \).

The solutions to the characteristic equation, often referred to as the roots, dictate the form of the homogeneous solution. Depending on the nature of these roots (real or complex), one can construct the general solution of the homogeneous equation, which is critical before adding the particular solution related to the non-homogeneous part.

Homogeneous Solution

The homogeneous solution, denoted as \( y_h(t) \), is the solution of a differential equation when the non-homogeneous term (\text like the forcing term) is set to zero. It represents the system's behavior in the absence of external influences. For second-order equations, the homogeneous solution typically includes terms like \( C_1\sin(\omega t) \) and \( C_2\cos(\omega t) \) where \( \omega \) is a constant and \( C_1, C_2 \) are arbitrary constants determined by initial conditions.

In our exercise, by finding the solution to the homogeneous equation \( y'' + \omega^2y=0 \) and applying the given initial conditions, we find the structure of the system's undisturbed response.

Particular Solution

The concept of a particular solution, \( y_p(t) \), deals with the inclusion of non-homogeneous terms in differential equations. Unlike the homogeneous solution that describes the system's natural dynamics, the particular solution specifically addresses the influence of the external, non-homogeneous term, in this case, \( \sin(nt) \).

To find it, one often assumes a form that is similar to the non-homogeneous term, but with unknown coefficients, and then determines these coefficients by plugging the assumed solution into the original equation. The full solution of the differential equation is the sum of the homogeneous and particular solutions, taking into account the contribution of both the inherent forces and the external forces. For the initial value problem we're examining, the particular solution \( y_p(t) \) shows how \( \sin(nt) \) influences the equation's behavior, ensuring a comprehensive understanding of the system's dynamics under periodic forcing.