Question

Consider the nonhomogeneous differential equation \(y^{\prime \prime}-y=e^{i 2 t}\). The complementary solution is \(y_{C}=c_{1} e^{t}+c_{2} e^{-t}\). Recall from Euler's formula that \(e^{i 2 t}=\cos 2 t+\) \(i \sin 2 t\). Therefore, the right-hand side is a (complex-valued) linear combination of functions for which the method of undetermined coefficients is applicable. (a) Assume a particular solution of the form \(y_{P}=A e^{i 2 t}\), where \(A\) is an undetermined (generally complex) coefficient. Substitute this trial form into the differential equation and determine the constant \(A\). (b) With the constant \(A\) as determined in part (a), write \(y_{P}(t)=A e^{i 2 t}\) in the form \(y_{P}(t)=u(t)+i v(t)\), where \(u(t)\) and \(v(t)\) are real- valued functions. (c) Show that \(u(t)\) and \(v(t)\) are themselves particular solutions of the following differential equations: $$ u^{\prime \prime}-u=\operatorname{Re}\left[e^{i 2 t}\right]=\cos 2 t \quad \text { and } \quad v^{\prime \prime}-v=\operatorname{Im}\left[e^{i 2 t}\right]=\sin 2 t $$ Therefore, the single computation with the complex-valued nonhomogeneous term yields particular solutions of the differential equation for the two real- valued nonhomogeneous terms forming its real and imaginary parts.

Short answer

Answer: The particular solutions for the real-valued nonhomogeneous terms are \(u(t) = -\frac{1}{5}\cos 2t\) and \(v(t) = -\frac{1}{5}\sin 2t\).

Step-by-step solution

Substitute trial form into the differential equation

Let's assume the trial form \(y_P = A e^{i 2 t}\) and calculate its second derivative: $$ y_P' = 2iAe^{i2t} $$ $$ y_P'' = - 4 A e^{i 2 t} $$ Now, substitute \(y_P\) and \(y_P''\) into the given differential equation: $$ y_P'' - y_P = - 4 A e^{i 2 t} - A e^{i 2 t} = e^{i 2 t} $$

Determine the constant A

In order to determine \(A\), we compare the coefficients of the exponential terms in the equation: $$-4A - A = 1$$ Solving for A, we get: $$ A = -\frac{1}{5} $$ So, the particular solution is: $$ y_P(t)=-\frac{1}{5}e^{i2t} $$

Separate real and imaginary parts

Now we write the particular solution in the form \(y_P(t)=u(t)+iv(t)\) using Euler's formula: $$ y_P(t)=-\frac{1}{5}( \cos 2t + i\sin 2t ) $$ Thus, we have \(u(t)=-\frac{1}{5}\cos 2t\) and \(v(t)=-\frac{1}{5}\sin 2t\).

Show that u(t) and v(t) are particular solutions

To show that \(u(t)\) and \(v(t)\) are particular solutions of the corresponding differential equations, we need to find their second derivatives and substitute them into the given differential equations. For \(u(t)\), we have: $$ u'(t) = \frac{2}{5}\sin 2t $$ $$ u''(t) = \frac{4}{5}\cos 2t $$ Substituting into the first given differential equation: $$ u''(t) - u(t) = \frac{4}{5}\cos 2t - \left(-\frac{1}{5}\cos 2t\right) = \cos 2t $$ For \(v(t)\), we have: $$ v'(t) = -\frac{2}{5}\cos 2t $$ $$ v''(t) = \frac{4}{5}\sin 2t $$ Substituting into the second given differential equation: $$ v''(t) - v(t) = \frac{4}{5}\sin 2t - \left(-\frac{1}{5}\sin 2t\right) = \sin 2t $$ Thus, \(u(t)\) and \(v(t)\) are indeed particular solutions of their respective differential equations.

Key concepts

Complementary Solution

When solving nonhomogeneous differential equations, identifying the complementary solution, or homogeneous solution, is a crucial step. The complementary solution, denoted as y_C, corresponds to the solution of the associated homogeneous equation—that is, the equation with zero on the nonhomogeneous (right) side.

For example, in the exercise y'' - y = e^{i 2t}, the complementary solution is found by setting the right-hand side to zero and solving y'' - y = 0. The given complementary solution y_C = c_1 e^t + c_2 e^{-t} involves constants c_1 and c_2, which are determined by the initial conditions of the problem. This solution represents the general response of the system devoid of any external forces.

Method of Undetermined Coefficients

The method of undetermined coefficients is a systematic way to find a particular solution to nonhomogeneous differential equations. This method works well when the nonhomogeneous term is a linear combination of certain types of functions, including exponentials, polynomials, sines, and cosines.

In the method, we assume a form for the particular solution y_P based on the nonhomogeneous term, bearing in mind the functions it consists of. We insert ‘undetermined’ coefficients into this assumed form, hence the name — for instance, y_P = A e^{i 2t} in the provided exercise. Afterward, we differentiate this form as needed and substitute back into the differential equation to solve for these unknown coefficients. It is critical to accurately predict the form of y_P to ensure that the process yields a valid particular solution.

Euler's Formula

Euler's formula, an essential tool in solving differential equations with complex numbers, reads e^{it} = \(\cos t + i \sin t\). It bridges real and complex analysis by representing complex exponentials through trigonometric functions. In our exercise, Euler's formula helps in converting the complex exponential function e^{i 2t} into its equivalent trigonometric form for further analysis.

This formula is invaluable not only to simply convey complex functions but also to facilitate the separation of the real and imaginary parts of solutions in differential equations. When the nonhomogeneous term involves e^{it}, as it does in our exercise, we can use this formula to express the particular solution in terms of real-valued functions, simplifying the interpretation and subsequent calculation of the original differential equation.

Particular Solutions

Particular solutions in the context of nonhomogeneous differential equations are specific solutions that satisfy the nonhomogeneous equation. They are not general solutions that contain arbitrary constants. Rather, they embody the influence of the nonhomogeneous term on the system represented by the equation.

In our problem, we derived particular solutions u(t) and v(t) for the real and imaginary parts, respectively. We've done so by separately applying the method of undetermined coefficients to each component after employing Euler's formula. The particular solutions reflect the actual response of the system when subject to the external force or input given by the nonhomogeneous term. The complete solution to the nonhomogeneous differential equation combines these particular solutions with the complementary solution, adapting to both the inherent properties of the system (represented by y_C) and the external influences (y_P).