Question

Write each of the functions in the form \(A e^{\omega t} \cos \beta t+i B e^{i t} \sin \beta t\), where \(\alpha, \beta, A\), and \(B\) are real numbers. (a) \(2 e^{i \sqrt{2} t}\) (b) \(\frac{2}{\pi} e^{-(2+3 i) t}\) (c) \(-\frac{1}{2} e^{2 t+i(t+\pi)}\) (d) \(\left(\sqrt{3} e^{(1+i) t}\right)^{3}\) (e) \(\left(-\frac{1}{\sqrt{2}} e^{i \pi t}\right)^{3}\)

Short answer

The expression in the desired form for the given function \(-\frac{1}{2} e^{3i \pi t}\) is \(\frac{-1}{2} \cos(3\pi t) - \frac{1}{2}i \sin(3\pi t)\).

Step-by-step solution

Identify coefficients

In this case, the given expression is \(2e^{i\sqrt{2}t}\), where \(A=2\), \(\omega = 0\), and \(\beta = \sqrt{2}\).

Apply Euler's Formula

Using Euler's formula, we get \[2 e^{i\sqrt{2}t} = 2(\cos(\sqrt{2}t) + i\sin(\sqrt{2}t))\] Therefore, the expression in the desired form is \[2 e^{0} \cos(\sqrt{2}t) + i 2 e^{0} \sin(\sqrt{2}t)\] (b) \(\frac{2}{\pi} e^{-(2+3 i) t}\)

Identify coefficients

In this case, the given expression is \(\frac{2}{\pi}e^{-(2+3i)t}\), where \(A=\frac{2}{\pi}\), \(\omega = -2\), and \(\beta = 3\).

Apply Euler's Formula

Using Euler's formula, we get \[\frac{2}{\pi} e^{-(2+3 i) t} = \frac{2}{\pi} e^{-2t} (\cos(3t) + i\sin(3t))\] Therefore, the expression in the desired form is \[\frac{2}{\pi} e^{-2t} \cos(3t) + i \frac{2}{\pi} e^{-2t} \sin(3t)\] (c) \(-\frac{1}{2} e^{2 t+i(t+\pi)}\)

Identify coefficients

In this case, the given expression is \(-\frac{1}{2}e^{2t+i(t+\pi)}\), where \(A=-\frac{1}{2}\), \(\omega = 2\), and \(\beta = 1\).

Apply Euler's Formula

Using Euler's formula, we get \[-\frac{1}{2} e^{2 t+i(t+\pi)} = -\frac{1}{2} e^{2t} (\cos(t+\pi) + i\sin(t+\pi))\] Therefore, the expression in the desired form is \[-\frac{1}{2} e^{2t} \cos(t+\pi) + i \left(-\frac{1}{2} e^{2t} \sin(t+\pi)\right)\] (d) \(\left(\sqrt{3} e^{(1+i) t}\right)^{3}\)

Simplify the expression

First, simplify the expression by expanding the power of 3: \[\left(\sqrt{3} e^{(1+i) t}\right)^{3} = 3 e^{(3+3i)t}\]

Identify coefficients

In this case, the simplified expression is \(3e^{(3+3i)t}\), where \(A=3\), \(\omega = 3\), and \(\beta = 3\).

Apply Euler's Formula

Using Euler's formula, we get \[3 e^{(3+3 i) t} = 3 e^{3t} (\cos(3t) + i\sin(3t))\] Therefore, the expression in the desired form is \[3 e^{3t} \cos(3t) + i 3 e^{3t} \sin(3t)\] (e) \(\left(-\frac{1}{\sqrt{2}} e^{i \pi t}\right)^{3}\)

Simplify the expression

First, simplify the expression by expanding the power of 3: \[\left(-\frac{1}{\sqrt{2}} e^{i \pi t}\right)^{3} = \frac{-1}{2} e^{3i\pi t}\]

Identify coefficients

In this case, the simplified expression is \(\frac{-1}{2} e^{3i \pi t}\), where \(A=-\frac{1}{2}\), \(\omega = 0\), and \(\beta = 3\pi\).

Apply Euler's Formula

Using Euler's formula, we get \[\frac{-1}{2} e^{3i \pi t} = \frac{-1}{2} (\cos(3\pi t) + i\sin(3\pi t))\] Therefore, the expression in the desired form is \[\frac{-1}{2} e^{0} \cos(3\pi t) + i \frac{-1}{2} e^{0} \sin(3\pi t)\]

Key concepts

Euler's Formula

Euler's formula, a fundamental cornerstone in the field of complex analysis, represents complex exponentials in terms of trigonometric functions. The beauty of this formula lies in its simplicity and power: \[ e^{ix} = \text{cos}(x) + i\text{sin}(x) \. \] With \(i\) being the imaginary unit and \(x\) a real number, this relationship allows us to move seamlessly between exponential and trigonometric representations of complex numbers.

This pivotal formula not only enables the solution of differential equations with complex coefficients but also greatly simplifies the analysis of oscillations and waves in physics. By understanding Euler's formula, students can see how complex exponentials model periodic phenomena and learn to rewrite complex functions in a more interpretable form, as demonstrated in the exercise solutions.

Differential Equations

Differential equations play a central role in modeling the behavior of various systems across disciplines such as physics, engineering, biology, and economics. They consist of equations involving an unknown function and its derivatives. The crux of solving a differential equation involves finding a function that satisfies the given relationship.

In the context of the exercise, by leveraging the transformative power of Euler's formula, we can express complex-valued solutions of linear differential equations in terms of real functions like \(\cos(\beta t)\) and \(\sin(\beta t)\), which are often easier to interpret and understand. Applying this to solve differential equations with complex coefficients greatly simplifies the process and aids in visualizing solutions.

Complex Numbers

Complex numbers extend the idea of one-dimensional number lines to a two-dimensional complex plane with real and imaginary axes. Every complex number can be written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^{2} = -1\).

In the provided solution examples, we see complex numbers being expressed in their exponential form. This not only compactly conveys the magnitude and phase of the complex numbers but also simplifies multiplication and division operations. Understanding the nature and behavior of complex numbers is central to interpreting solutions to equations that describe phenomena such as electrical currents, signal processing, and quantum mechanics.

Boundary Value Problems

Boundary value problems are a specific type of differential equation where we look for solutions that satisfy certain conditions at the boundaries of the domain. In physical terms, these boundaries can represent the edge of a string, the surface of a drum, or the terminals of an electrical component.

While not explicitly covered in the example exercise, boundary value problems often require complex solutions, particularly in systems governed by wave equations or in quantum mechanics. The ability to express these solutions using complex exponentials greatly facilitates the process of meeting the boundary conditions and finding meaningful solutions to physical problems.