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Chapter 3: Problem 31

Find the solution of the differential equation that satisfies the given conditions. $$ y^{\prime \prime \prime}+y=\cos t, \quad y(0)=1, \quad|y(t)| \leq 2 \text { for all } t, \quad 0 \leq t<\infty $$

Short Answer

Expert verified
In summary, we analyzed the given third-order linear inhomogeneous differential equation and followed the steps to find the complementary and particular solutions. We used the initial condition and checked if the general solution satisfies the given limit condition. The general solution is: $$ y(t) = y_c(t) + y_p(t) = C_1e^{m_1t} + C_2e^{m_2t} + C_3e^{m_3t} + \cos t $$ with the constants \(C_1\), \(C_2\), and \(C_3\) determined by additional initial conditions, if provided. The solution satisfies the given limit condition of \(|y(t)| \leq 2\) for all \(t\), where \(0 \leq t < \infty\).

Step by step solution

01

Identify the differential equation type

The given differential equation is: $$ y^{\prime \prime \prime} + y = \cos t $$ The equation is a third-order linear inhomogeneous differential equation because it contains the third derivative and a non-zero term (cosine) on the right-hand side.
02

Find the complementary function

To find the complementary function, we consider the homogeneous version of the given differential equation: $$ y^{\prime \prime \prime} + y = 0 $$ We use the auxiliary equation for this homogeneous differential equation: $$ m^3 + 1 = 0 $$ This equation has three complex solutions, m1, m2, and m3. Then, the complementary function can be presented as: $$ y_c(t) = C_1e^{m_1t} + C_2e^{m_2t} + C_3e^{m_3t} $$ where \(C_1\), \(C_2\), and \(C_3\) are constants to be determined later.
03

Find the particular solution

To find the particular solution, we use the undetermined coefficients method. We guess a solution of the form: $$ y_p(t) = A\cos t + B\sin t $$ where A and B are constants to be determined. Now, compute the first, second, and third derivatives of \(y_p(t)\): $$ y_p^{\prime}(t) = -A\sin t + B\cos t \\ y_p^{\prime \prime}(t) = -A\cos t - B\sin t \\ y_p^{\prime \prime \prime}(t) = A\sin t - B\cos t $$ Substituting these derivatives into the given differential equation: $$ (A\sin t - B\cos t) + (A\cos t + B\sin t) = \cos t $$ It yields two equations to solve for A and B: $$ A\cos t + B\sin t = \cos t \\ A\sin t - B\cos t = 0 $$ Upon solving these equations, we get \(A=1\) and \(B=0\). Therefore, the particular solution is: $$ y_p(t) = \cos t $$
04

Use the initial conditions to determine the constants

Now, we combine the complementary and particular solutions to get the general solution: $$ y(t) = y_c(t) + y_p(t) = C_1e^{m_1t} + C_2e^{m_2t} + C_3e^{m_3t} + \cos t $$ The initial condition given is \(y(0) = 1\). We substitute \(t=0\) into the general solution: $$ y(0) = C_1e^{m_10} + C_2e^{m_20} + C_3e^{m_30} + \cos 0 = 1 $$ Since \(e^{m_1t}\), \(e^{m_2t}\), and \(e^{m_3t}\) are linearly independent, we can conclude that: $$ C_1 + C_2 + C_3 + 1 = 1 $$ Thus, we have \(C_1\), \(C_2\), and \(C_3\). These constants can be determined if additional information about initial conditions is given.
05

Check the solution satisfies the given limit condition

The given condition is: $$ |y(t)| \leq 2 \text{ for all } t, \quad 0 \leq t < \infty $$ To verify if the general solution satisfies this condition, consider the exponential term in the general solution. Since it involves imaginary exponents, it is oscillatory in nature and bounded within a certain range. We also have the trigonometric term, the cosine function, which is also bounded between -1 and 1. Thus, we can conclude that the general solution satisfies the given condition in the given range of \(t\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Third-Order Linear Inhomogeneous Differential Equation
A third-order linear inhomogeneous differential equation features derivatives up to the third order and includes a non-zero function on the right-hand side, unlike homogeneous equations which equate to zero. An example of such an equation is:
\[ y^{\prime \prime \prime} + y = \cos t \]
Solving these kinds of equations often requires a two-step approach. Firstly, the solution to the corresponding homogeneous equation (where the right-hand side is zero) is found. Secondly, a particular solution to the non-homogeneous equation is determined. The sum of these two components yields the general solution to the inhomogeneous equation. This comprehensive approach ensures that all possible behaviors of the system described by the equation are captured.
Complementary Function
The complementary function, denoted by \( y_c(t) \), is the solution to the homogeneous part of the differential equation and contains a number of arbitrary constants equal to the order of the equation. For the example:
\[ y^{\prime \prime \prime} + y = 0 \]
we find the complementary function using the roots of the auxiliary equation associated with the differential equation. The roots dictate the form of \( y_c(t) \), which could include exponential, sine, and cosine functions, depending on whether the roots are real or complex. As part of the overall solution, the complementary function captures the system's natural behavior, free from external influences.
Particular Solution
In contrast to the complementary function, the particular solution, denoted as \( y_p(t) \), specifically addresses the inhomogeneous portion of the differential equation. Using a strategy appropriate to the non-zero function on the right-hand side, such as the undetermined coefficients method, we guess a form for \( y_p(t) \) that resembles the inhomogeneous part. In our example, a solution of the form
\[ y_p(t) = A\cos t + B\sin t \]
is hypothesized. Once substituted back into the original equation, this form allows us to solve for the coefficients, yielding a solution that reflects the impact of the external force or input described by the non-homogeneous term.
Undetermined Coefficients Method
The undetermined coefficients method is a technique used to find the particular solution for linear inhomogeneous differential equations, especially when the non-homogeneous term is a simple polynomial, exponential, sine, or cosine function. The procedure involves guessing a form for the particular solution with undefined coefficients and then differentiating it as required. By substituting these derivatives back into the original equation, the undetermined coefficients can be found through comparing coefficients, allowing for a specific solution to be derived that fits alongside the complementary function.
Initial Conditions
Initial conditions are specific values given for the function and its derivatives at a particular point. These conditions allow us to uniquely determine the constants in the complementary function. For a third-order differential equation, we typically need three conditions to solve for three constants. For example, in our exercise, only one initial condition is provided:
\[ y(0) = 1 \]
Without additional conditions, we cannot find the precise values for all constants. Nonetheless, initial conditions play a crucial role in anchoring the general solution to a particular scenario, ensuring the result fits the context of the problem. After applying the initial conditions, the final step is to check whether the general solution satisfies any given constraints, such as the inequality \( |y(t)| \leq 2 \). This ensures the solution's validity over the specified range of \( t \).

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Most popular questions from this chapter

A spring-mass-dashpot system consists of a \(10-\mathrm{kg}\) mass attached to a spring with spring constant \(k=100 \mathrm{~N} / \mathrm{m}\); the dashpot has damping constant \(7 \mathrm{~kg} / \mathrm{s}\). At time \(t=0\), the system is set into motion by pulling the mass down \(0.5 \mathrm{~m}\) from its equilibrium rest position while simultaneously giving it an initial downward velocity of \(1 \mathrm{~m} / \mathrm{s}\). (a) State the initial value problem to be solved for \(y(t)\), the displacement from equilibrium (in meters) measured positive in the downward direction. Give numerical values to all constants involved. (b) Solve the initial value problem. What is \(\lim _{t \rightarrow \infty} y(t)\) ? Explain why your answer for this limit makes sense from a physical perspective. (c) Plot your solution on a time interval long enough to determine how long it takes for the magnitude of the vibrations to be reduced to \(0.1 \mathrm{~m}\). In other words, estimate the smallest time, \(\tau\), for which \(|y(t)| \leq 0.1 m, \tau \leq t<\infty\).

(a) Obtain the general solution of the differential equation. (b) Impose the initial conditions to obtain the unique solution of the initial value problem. (c) Describe the behavior of the solution as \(t \rightarrow-\infty\) and \(t \rightarrow \infty\). In each case, does \(y(t)\) approach \(-\infty,+\infty\), or a finite limit? $$ 16 y^{\prime \prime}-8 y^{\prime}+y=0, \quad y(0)=-4, \quad y^{\prime}(0)=3 $$

Consider the initial value problem $$ m y^{\prime \prime}+\gamma y^{\prime}+k y=F(t), \quad y(0)=0, \quad y^{\prime}(0)=0, $$ modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force \(F(t)\), where the unit of force is the newton (N). Assume that \(m=2 \mathrm{~kg}, \gamma=8 \mathrm{~kg} / \mathrm{s}\), and \(k=80 \mathrm{~N} / \mathrm{m} .\) (a) Solve the initial value problem for the given applied force. In Exercise 10 , use the fact that the system displacement \(y(t)\) and velocity \(y^{\prime}(t)\) remain continuous at times when the applied force is discontinuous. (b) Determine the long-time behavior of the system. In particular, is \(\lim _{t \rightarrow \infty} y(t)=0\) ? If not, describe in qualitative terms what the system is doing as \(t \rightarrow \infty\). $$ F(t)=20 \cos 8 t $$

A spring and dashpot system is to be designed for a \(32-1 b\) weight so that the overall system is critically damped. (a) How must the damping constant \(\gamma\) and spring constant \(k\) be related? (b) Assume the system is to be designed so that the mass, when given an initial velocity of \(4 \mathrm{ft} / \mathrm{sec}\) from its rest position, will have a maximum displacement of 6 in. What values of damping constant \(\gamma\) and spring constant \(k\) are required?

In each exercise, (a) Find the general solution of the differential equation. (b) If initial conditions are specified, solve the initial value problem. $$ y^{\prime \prime \prime}+y^{\prime \prime}+4 y^{\prime}+4 y=0 $$
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