Question

For each differential equation, (a) Find the complementary solution. (b) Formulate the appropriate form for the particular solution suggested by the method of undetermined coefficients. You need not evaluate the undetermined coefficients. $$ y^{(4)}-y=t e^{-t}+(3 t+4) \cos t $$

Short answer

Based on the given information: (a) The complementary solution is: $$ y_c(t) = C_1 e^t + C_2 e^{-t} + C_3 \cos t + C_4 \sin t $$ (b) The appropriate form for the particular solution suggested by the method of undetermined coefficients is: $$ y_p(t) = Ate^{-t} + Be^{-t} + (Ct + D) \cos t + (Et + F) \sin t $$

Step-by-step solution

Find the complementary solution

To find the complementary solution, we consider the corresponding homogeneous equation: $$ y^{(4)} - y = 0 $$ We can find the characteristic equation by making the substitution \(y=e^{rt}\), which gives: $$ r^4 - 1 = 0 $$ This equation can be factored as \((r^2 + 1)(r^2 - 1) = (r^2 + 1)(r - 1)(r + 1) = 0\). Thus the roots are \(r_1 = 1, r_2 = -1, r_3 = i, r_4 = -i\). Therefore, the complementary solution is given by: $$ y_c(t) = C_1 e^t + C_2 e^{-t} + C_3 \cos t + C_4 \sin t $$ where \(C_1, C_2, C_3\) and \(C_4\) are constants.

Formulate the appropriate form for the particular solution

Now we need to find an appropriate form for the particular solution using the method of undetermined coefficients. The forcing function in our given differential equation is \(te^{-t} + (3t + 4) \cos t\). As \(te^{-t}\) and \((3t + 4) \cos t\) are linearly independent, we can find two particular solutions for each term and sum them up.

Step 2a: Finding the appropriate form for the particular solution related to the term \(te^{-t}\)

As \(te^{-t}\) doesn't resemble any of the terms in the complementary solution, we assume the following form for the particular solution related to \(te^{-t}\): $$ y_{p1}(t) = Ate^{-t} + Be^{-t} $$ Where \(A\) and \(B\) are constants to be determined.

Step 2b: Finding the appropriate form for the particular solution related to the term \((3t + 4) \cos t\)

Considering the term \((3t + 4) \cos t\), we notice it has a cosine term, which is also part of our complementary solutions. Taking this into account, we can assume the following form for the particular solution related to \((3t + 4) \cos t\): $$ y_{p2}(t) = (Ct + D) \cos t + (Et + F) \sin t $$ Where \(C, D, E\), and \(F\) are constants to be determined.

Combine the particular solutions

Combining both particular solutions found in steps 2a and 2b, we have: $$ y_p(t) = y_{p1}(t) + y_{p2}(t) = Ate^{-t} + Be^{-t} + (Ct + D) \cos t + (Et + F) \sin t $$ Where \(A, B, C, D, E\), and \(F\) are constants to be determined. In this problem, we are not asked to determine the undetermined coefficients. However, if needed, we would take the derivatives of the particular solution and substitute them into the given differential equation, then solve for the constants.

Key concepts

Complementary Solution

When dealing with differential equations, especially higher-order ones, the complementary solution is a key concept. The complementary solution, denoted as \( y_c(t) \), is related to the homogeneous part of the equation. In this context, a homogeneous equation means that the equation is equal to zero. For example, for the given problem, the homogeneous equation is \( y^{(4)} - y = 0 \).Finding the complementary solution involves solving a characteristic equation, which results from substituting \( y=e^{rt} \) into the homogeneous equation. For our exercise, this substitution yields: \( r^4 - 1 = 0 \). Solving the characteristic equation helps us find the roots: \( r_1 = 1, \) \( r_2 = -1, \) \( r_3 = i, \) and \( r_4 = -i \). Each real root corresponds to an exponential function in the solution, and each complex pair results in terms involving sine and cosine. Thus, \( y_c(t) = C_1 e^t + C_2 e^{-t} + C_3 \cos t + C_4 \sin t \), where \( C_1, C_2, C_3, \) and \( C_4 \) are constants determined by initial conditions.Understanding the complementary solution is crucial as it represents behaviors of the solution related to the system's natural response, without external forces.

Particular Solution

The particular solution, denoted as \( y_p(t) \), addresses the non-homogeneous part of a differential equation. It accounts for the effect of the non-zero terms in the equation. In the given problem, these terms are \( t e^{-t} +(3t + 4) \cos t \).While the complementary solution represents the inherent dynamics of the system, the particular solution reflects how the system responds to external forces. To find the particular solution, it is often essential to use the method of undetermined coefficients. This method involves guessing a form for \( y_p(t) \) that mirrors the external forces and then determining the unknown coefficients.For \( t e^{-t} \), there's no overlap with any terms from the complementary solution, suggesting a trial form of \( y_{p1}(t) = Ate^{-t} + Be^{-t} \). Meanwhile, for \( (3t + 4) \cos t \), the trial form needs to account for possible overlapping terms in the complementary solution, resulting in \( y_{p2}(t) = (Ct + D) \cos t + (Et + F) \sin t \).The actual process of determining these coefficients requires substituting \( y(t) = y_c(t) + y_p(t) \) back into the original differential equation to solve for constants \( A, B, C, D, E, \) and \( F \). This part is left as an exercise unless specified otherwise.

Method of Undetermined Coefficients

The method of undetermined coefficients offers a systematic approach to finding particular solutions of linear differential equations with constant coefficients. The main idea is to assume a form for the particular solution \( y_p(t) \) that mimics the non-homogeneous part of the given equation. This method relies on recognizing patterns and adjusting the assumed form if necessary.Typically, the process involves:

• Identifying the non-homogeneous term of the differential equation.
• Assuming a trial solution with undetermined coefficients that appropriately reflects the structure of the non-homogeneous term. Adjust the form if overlaps with the complementary solution occur.
• Substituting the trial solution into the differential equation.
• Solving for the unknown coefficients by ensuring the trial solution satisfies the equation.

This method is efficient for differential equations where non-homogeneous terms are polynomials, exponentials, sines, and cosines, or a combination thereof.For our problem, two forms are assumed: one for \( t e^{-t} \) and another for \( (3t + 4) \cos t \). Each is designed to suit the individual characteristics of these terms. Hence, we form a combined particular solution that sums up both contributions.

Homogeneous Equation

A homogeneous equation is a type of differential equation in which the terms of the equation equal zero. In contrast to non-homogeneous equations, homogeneous equations do not have any forcing terms or free components driving the change in the system. The function's behavior is entirely dictated by the initial conditions and its intrinsic dynamics.In the context of the exercise, the homogeneous equation is \( y^{(4)} - y = 0 \). Solving it means finding a complementary solution, which describes how the system behaves inherently, assuming no external influences.Homogeneous equations are foundational because understanding them helps in building solutions to non-homogeneous equations. Often, solutions for differential equations are constructed by combining solutions to the associated homogeneous problem with particular solutions to the non-homogeneous problem.The characteristic equation derived from a homogeneous equation gives us the necessary roots, which form the basis for the complementary solution. This complementary solution encapsulates the natural response of the system to initial conditions.