Question

Rewrite the function \(y(t)\) in the form \(y(t)=R e^{a t} \cos (\beta t-\delta)\), where \(0 \leq \delta<2 \pi\). Use this representation to sketch a graph of the given function, on a domain sufficiently large to display its main features. $$ y(t)=e^{t} \cos t+\sqrt{3} e^{t} \sin t $$

Short answer

So we have: $$ y(t) = e^t (\cos t + \sqrt{3} \sin t) \\ = e^t [(\cos t)(\cos 0) + (\sqrt{3} \sin t)(\cos 60^{\circ}) + (\sqrt{3} \sin t)(\sin 60^{\circ}) - (\cos t)(\sin 0)] \\ = e^t \cos (60^{\circ})(\cos t + \sqrt{3} \sin t) $$ Now we can use the trigonometric identity in reverse: $$ y(t) = e^t \cos (60^{\circ} - t) $$ We can rewrite this in the desired form by dividing both sides by \(\cos(60^{\circ})\): $$ y(t) = \frac{e^t}{\cos(60^{\circ})} \cos (-t + 60^{\circ}) = Re^{at} \cos (\beta t - \delta) $$ Comparing both sides, we can identify \(R\), \(a\), \(\beta\), and \(\delta\). #tag_title# Step 2: Identify the values of \(R\), \(a\), \(b\), and \(\delta\) #tag_content# From the rewritten function, we can see that: $$ R = \frac{1}{\cos(60^{\circ})}, \quad a = 1, \quad \beta = -1, \quad \delta = 60^{\circ} $$ Now that we've identified the values for \(R\), \(a\), \(\beta\), and \(\delta\), we can move on to sketching the graph of y(t) on a domain large enough to display its main features. #tag_title# Step 3: Create and describe the graph of y(t) #tag_content# To sketch the graph of y(t), we'll consider the following features: 1. The amplitude, given by \(R = \frac{1}{\cos(60^{\circ})}\) 2. The phase shift, given by \(\delta = 60^{\circ}\) 3. The period, which is given by \(\frac{2\pi}{\vert\beta\vert} = 2\pi\) 4. The growth factor, given by the exponential term \(e^{at} = e^t\) As \(t\) increases, the function y(t) grows in amplitude due to the growth factor \(e^t\). This creates an oscillating function with increasing amplitude. It oscillates with a period of \(2\pi\) and has a phase shift of \(60^{\circ}\). Following these observations, you should sketch a graph of y(t), displaying the increasing amplitude of the oscillations and the phase shift. You can use a range of -2π ≤ t ≤ 2π to capture the essential features of the graph.

Step-by-step solution

Rewrite the given function in the desired form

Let's start with the given function: $$ y(t) = e^t \cos t + \sqrt{3} e^t \sin t $$ Now, we want to rewrite this function in the form of \(y(t) = R e^{a t} \cos (\beta t - \delta)\). To achieve this, we will factor out \(e^t\) from both terms and use the trigonometric identity \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\). -*-

Key concepts

Differential Equations

Differential equations are a class of equations that involve rates of change and the functions themselves. They are a fundamental tool in expressing the relationships between a function and its derivatives. For example, if we have a function representing the position of a particle over time, a differential equation might describe its speed and acceleration at any point.

In the context of complex numbers and trigonometric forms, they can be used to model oscillations and waves, such as the motion of a pendulum or electrical currents in circuits. The goal is to find a function that satisfies the given differential equation, which may embody a physical or geometric phenomenon.

The original exercise involves a particular form of a differential equation solution where the function is exponetially multiplied by a trigonometric cosine function. These kinds of solutions often emerge from solving second-order linear differential equations common in physics and engineering.

Euler's Formula

Euler's formula is a cornerstone of complex analysis and can bridge trigonometry with complex numbers. It states that for any real number x, \( e^{ix} = \cos x + i \sin x \). This elegant formula connects the exponential function to the trigonometric functions, cos and sin, via the imaginary unit \(i\).

In the step-by-step solution, Euler's formula is key to rewriting the given function in trigonometric form. By recognizing the components \( \cos t \) and \( \sin t \) within the original equation, and that they can be represented as the real and imaginary parts of \( e^{it} \) respectively, we can combine and manipulate these elements to rewrite the equation in the desired format involving \( \cos(\beta t - \delta)\).

Harmonic Motion

Defining Harmonic Motion

Harmonic motion refers to a pattern of motion that repeats itself in a regular cycle, such as the swinging of a pendulum or the vibration of a guitar string. It's described as \( y(t) = A \cos(\omega t + \varphi) \), where \( A \) is the amplitude, \( \omega \) the angular frequency, and \( \varphi \) the phase shift.

The motion is called \( harmonic \) because it embodies a perfect rhythm, one of the central notions in music, but in physics and mathematics, it also represents one of the simplest forms of repetitive motion. In the given exercise, the function \( y(t) = e^t \cos t + \sqrt{3} e^t \sin t \) represents an exponentially damped harmonic oscillation, which exhibits a pattern of decay often seen in nature and engineering systems.

Boundary Value Problems

Boundary value problems (BVPs) are problems in which a differential equation is combined with a set of additional restraints, called boundary conditions. These extra conditions are specified at the boundaries of the domain over which the equation is defined.

For example, the temperature distribution in a rod over time may be modeled by a differential equation with boundary conditions that determine the temperature at each end of the rod.

In our problem, although it is not directly a BVP, the principle of using constraints to describe a system is still applicable. We're given a function and a requirement to fit it into a specific form that obeys a range constraint for the phase \( \delta \) which acts like a boundary condition for the allowed phase shift in the trigonometric representation of the complex number.