Question

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

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

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.

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.

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: $$ y_p(t) = \cos t $$

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.

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 \(t\).

Key concepts

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 \( t \).