Problem 20 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each of the functions below, determine whether the function is a solution to differential equation (i), differential equation (ii), or neither. Differential equations (i) and (ii) are given below. \(\begin{array}{ll}\text { i. } y^{\prime \prime}=16 y & \text { ii. } y^{\prime \prime}=-16 y\end{array}\) (a) \(y_{1}(t)=\sin 16 t\) (b) \(y_{2}(t)=e^{4 t}\) (c) \(y_{3}(t)=3 \cos 4 t\) (d) \(y_{4}(t)=\sin 4 t+1\) (e) \(y_{5}(t)=e^{-16 t}\) (f) \(y_{6}(t)=-3 e^{-4 t}\) (g) \(y_{7}(t)=e^{4 t}+3\) (h) \(y_{8}(t)=-\sin 4 t\)

Step-by-Step Solution

Answer

The function \(y_{2}(t)\) is a solution to differential equation i, \(y_{8}(t)\) is a solution to differential equation ii, and the others are solutions to neither.

Step 1: For \(y_{1}(t)=\sin 16 t\)

The second derivative of \(y_{1}(t)\) is \(-256 \sin 16 t\). This is neither equal to 16 times \(\sin 16 t\) nor -16 times \(\sin 16 t\), hence \(y_{1}(t)\) is a solution to neither (i) nor (ii).

Step 2: For \(y_{2}(t)=e^{4 t}\)

The second derivative of \(y_{2}(t)\) is \(16 e^{4 t}\). This is equal to 16 times \(e^{4 t}\) which implies \(y_{2}(t)\) is a solution to (i) but not (ii).

Step 3: For \(y_{3}(t)=3 \cos 4 t\)

The second derivative of \(y_{3}(t)\) is \(-48 \cos 4 t\). This is neither equal to 16 times \(3 \cos 4 t\) nor -16 times \(3 \cos 4 t\), hence \(y_{3}(t)\) is a solution to neither (i) nor (ii).

Step 4: For \(y_{4}(t)=\sin 4 t+1\)

The second derivative of \(y_{4}(t)\) is \(-16 \sin 4 t\). This is neither equal to 16 times \(\sin 4 t + 1\) nor -16 times \(\sin 4 t + 1\), hence \(y_{4}(t)\) is a solution to neither (i) nor (ii).

Step 5: For \(y_{5}(t)=e^{-16 t}\)

The second derivative of \(y_{5}(t)\) is \(256 e^{-16 t}\). This is neither equal to 16 times \(e^{-16 t}\) nor -16 times \(e^{-16 t}\), hence \(y_{5}(t)\) is a solution to neither (i) nor (ii).

Step 6: For \(y_{6}(t)=-3 e^{-4 t}\)

The second derivative of \(y_{6}(t)\) is \(48 e^{-4 t}\). This is neither equal to 16 times \(-3 e^{-4 t}\) nor -16 times \(-3 e^{-4 t}\), hence \(y_{6}(t)\) is a solution to neither (i) nor (ii).

Step 7: For \(y_{7}(t)=e^{4 t}+3\)

The second derivative of \(y_{7}(t)\) is \(16 e^{4 t}\). This is neither equal to 16 times \(e^{4 t} + 3\) nor -16 times \(e^{4 t} + 3\), hence \(y_{7}(t)\) is a solution to neither (i) nor (ii).

Step 8: For \(y_{8}(t)=-\sin 4 t\)

The second derivative of \(y_{8}(t)\) is \(16 \sin 4 t\). This is equal to -16 times \(-\sin 4 t\), which implies \(y_{8}(t)\) is a solution to (ii) but not (i).

Key Concepts

Second Derivative

In differential calculus, the second derivative provides information about the curvature of a function's graph and its acceleration. For any given function, the second derivative, denoted as \(y''\), is the derivative of its first derivative \(y'\). Calculating the second derivative can help us understand the concavity or convexity of a function in addition to identifying the points of inflection where the curvature changes direction.

If the second derivative is positive, the function is concave up, resembling a cup. If negative, the function is concave down, resembling a cap. In solving differential equations, finding the second derivative is crucial as it helps to see if a given function satisfies certain conditions or equations.

When given a differential equation such as \(y^{\prime \prime}=16y\), the solution involves finding a function whose second derivative equals 16 times the function itself. These differential equations often arise in the study of natural phenomena, engineering, and physics.

Solutions to Differential Equations

A solution to a differential equation is a function that satisfies the equation when substituted into it. Differential equations describe a relationship between a function and its derivatives and have numerous applications.

There are different types of differential equations. Here, we are dealing with second-order ordinary differential equations such as \(y^{\prime \prime}=16y\) or \(y^{\prime \prime}=-16y\). Each of these stands for a mathematical rule linking a function to its second derivative.

To check if a given function is a solution, we substitute the function and its derivatives into the equation and verify if the equation holds. Consideration of initial conditions or boundary values can also lead to unique solutions when tackling real-world problems.

In our exercise, understanding if a function like \(e^{4t}\) satisfies the equation helps us know if it correctly models the underlying phenomena, such as growth or oscillations.

Trigonometric Functions

Trigonometric functions like \(\sin(t)\) and \(\cos(t)\) are fundamental in the study of waves and oscillations. Regularly occurring in differential equations, they describe how a point moves around a circle at constant speed.

When dealing with differential equations such as \(y^{\prime \prime}=-16y\), trigonometric functions often serve as solutions. This is due to their properties after differentiating twice. For instance, the second derivative of \(\sin(4t)\) is \(-16\sin(4t)\), satisfying the given equation.

Understanding these functions' basic behaviors and properties, such as their periodic nature, amplitude, and phase, becomes essential especially in physics and engineering where oscillatory motions are prevalent. Trigonometric solutions are everywhere, from describing light waves to sound vibrations.

Exponential Functions

Exponential functions, like \(e^{kt}\), are key players in differential equations due to their property of always returning a constant multiple of themselves when differentiated. This makes them ideal for representing growth processes or radioactive decay.

In the context of our differential equation \(y^{\prime \prime}=16y\), the exponential function \(e^{4t}\) was shown to be a solution because its second derivative matches \(16e^{4t}\), fitting perfectly into the equation. Such behavior is linked to their innate qualities when modeled by constant rates.

They serve a vital role in both pure and applied mathematics, routinely appearing in compound interest calculations, population growth models, and signal processing. Their unique feature of maintaining their shape under differentiation makes them indispensable in a plethora of scientific contexts.