Problem 8 - Differential Equations and Linear Algebra

Question

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=e^{-t} \sin 2 t, \quad a=\pi / 6$$.

Step-by-Step Solution

Answer

The short answer to the question is: \[f(t-a) = e^{-t} \cdot e^{\pi/6} \sin (2t - \pi/3)\]

Step 1: Replace t with (t-a)

Replace every occurrence of the variable \(t\) with \((t-a)\) in the given function, where \(a = \pi/6\): \[f(t-a) = e^{-(t-a)} \sin 2(t-a)\] Step 2: Simplify the expression

Step 2: Simplify the exponent in the exponential term

Simplify the exponent of the exponential term in the function: \[e^{-(t-a)} = e^{-t + a} = e^{-t} \cdot e^a\] Remembering that \(a = \pi/6\), the expression can be written as: \[e^{-t} \cdot e^{\pi/6}\]

Step 3: Simplify the argument in the sin term

Simplify the argument of the sin term in the function: \[\sin 2(t-a) = \sin 2(t-\pi/6) = \sin (2t - 2\pi/6) = \sin (2t - \pi/3)\] Step 3: Combine the simplified terms

Step 4: Combine the simplified exponential and sin terms

Combine the simplified exponential and sin terms to obtain the final expression for \(f(t-a)\): \[f(t-a) = e^{-t} \cdot e^{\pi/6} \sin (2t - \pi/3)\]

Key Concepts

Time Shifting

When we talk about "Time Shifting," we are referring to adjusting the input of a function by a constant value.
This concept is commonly used in physics and engineering, especially with regard to signal processing.

• The basic idea is to slide or "shift" the function along the time axis.
• In this context, if you have a function like \( f(t) \), a time-shifted version \( f(t-a) \) means the function happens "\( a \)" units later (or earlier if \( a \) is negative).

For example, if you wanted to time shift the function \( f(t) = e^{-t} \sin 2t \) by \( a = \pi/6 \), each \( t \) in the equation would be replaced by \( t - \pi/6 \).
This plays an essential role when dealing with Laplace transforms as it helps analyze different time-based changes to functions.

Exponential Functions

Exponential functions are fundamentally structured as a constant raised to a variable power, like \( e^{-t} \).
These functions describe processes that increase or decrease rapidly; a classic example includes radioactive decay and population growth.

• The key characteristic of an exponential function is its constant rate of change.
• The base \( e \) (approximately 2.718) is a natural constant commonly used in these functions.

When the task requires to simplify \( e^{-(t-a)} \), we rewrite it as \( e^{-t + a} \), then further broken down to \( e^{-t} \cdot e^{a} \).
This transformation maintains the original behavior while accounting for the shift \( a \); when \( a = \pi/6 \), the term \( e^{\pi/6} \) becomes a multiplicative constant that will affect the function's amplitude.

Trigonometric Functions

Trigonometric functions like \( \sin \, \) and \( \cos \, \) describe the relations of angles on a unit circle.
They are periodic, meaning their values repeat at regular intervals, which is why they fit well into wave-like behavior in physics.

• They provide insights into oscillations, signals, and other waveforms.
• The argument of these functions, such as \( 2(t-a) \) in the expression \( \sin 2(t-a) \), represents a shifted angle in the trigonometric function.

When simplifying \( \sin 2(t-\pi/6) \), we focus on adjusting the angle due to the time shift.
You first rewrite it as \( \sin(2t - \pi/3) \), achieving a change in the function's phase but not its periodicity.
This modification affects where the cycle starts, effectively altering the position of the wave without changing its shape.