/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Pick out which functions are sol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 25

Pick out which functions are solutions to which differential equations. (Note: Functions may be solutions to more than one equation or to none; an equation may have more than one solution.) (a) \(\frac{d y}{d x}=-2 y\) (b) \(\frac{d y}{d x}=2 y\) (c) \(\frac{d^{2} y}{d x^{2}}=4 y\) (d) \(\frac{d^{2} y}{d x^{2}}=-4 y\) (I) \(y=2 \sin x\) (II) \(y=\sin 2 x\) (III) \(y=e^{2 x}\) (IV) \(y=e^{-2 x}\)

Short Answer

Expert verified
(a) IV; (b) III; (c) III; (d) I, IV.

Step by step solution

01

Check y=2sin(x) for each differential equation

For differential equation (a) \( \frac{d y}{d x} = -2y \), calculate the derivative \( \frac{d}{dx}(2\sin{x}) = 2\cos{x} \). Substituting back, we get \( 2\cos{x} = -2(2\sin{x}) \), which is False. Repeat for other equations. Only (d) \( \frac{d^2 y}{d x^2} = -4y \) with \( \frac{d^2}{dx^2} (2\sin{x}) = -2\sin{x} \) holds.
02

Check y=sin(2x) for each differential equation

For differential equation (a), calculate \( \frac{d}{dx}(\sin{2x}) = 2\cos{2x} \). Substituting back, we get \( 2\cos{2x} = -2(\sin{2x}) \), which is False. Check other equations similarly. None satisfy the given equations.
03

Check y=e^{2x} for each differential equation

For differential equation (b) \( \frac{d y}{d x} = 2 y \), calculate \( \frac{d}{dx}(e^{2x}) = 2e^{2x} \). Substituting back, we find \( 2e^{2x} = 2e^{2x} \), which is True. Also check (c) \( \frac{d^2 y}{d x^2} = 4 y \). Both equations work with this function.
04

Check y=e^{-2x} for each differential equation

For differential equation (a), \( \frac{d}{dx}(e^{-2x}) = -2e^{-2x} \), which satisfies \( \frac{d y}{d x} = -2y \). Also check equation (d) \( \frac{d^2 y}{d x^2} = -4 y \). Both equations hold

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solutions to Differential Equations
Differential equations describe relationships between functions and their derivatives, and solutions to these equations are functions that satisfy these relationships. To check if a function is a solution to a particular differential equation, you need to plug it into the equation and see if equality holds.
Typically, the process involves taking derivatives of the proposed solution function and substituting them back into the differential equation. If both sides of the equation are equal after substitution, the function can be considered a solution.
  • For example, if we have the differential equation \( \frac{d y}{d x} = 2 y \) and a candidate solution \( y = e^{2x} \), by differentiating and substituting, we verify if the function is valid.
  • In problem-solving, multiple functions might satisfy a single differential equation, and each function can often be a solution to more than one equation.
Second-Order Differential Equations
A second-order differential equation involves the second derivative of a function. Such equations are critically important in modeling various physical systems, including motion, waves, and many more phenomena.
To check a function against a second-order differential equation, calculate both the first and second derivatives, then substitute into the equation.
For instance, given the second-order differential equation \( \frac{d^2 y}{d x^2} = -4 y \), to confirm that \( y = 2 \sin x \) is a solution, compute the second derivative and substitute back, checking if both sides match.
  • This principal function showcases how solutions might involve sinusoidal (trigonometric) functions when the equation itself has a harmonic form.
  • Second-order differential equations are often solved using characteristic equations, particularly when they have constant coefficients.
Exponential Functions
Exponential functions are frequently encountered in the solutions of differential equations due to their distinct property of self-derivative resemblance. Specifically, functions like \( e^{kx} \) (where \( k \) is a constant) maintain their form even after differentiation.
In the domain of differential equations, this property makes them ideal candidates for solutions involving both growth and decay processes.
  • An example includes solving \( \frac{d y}{d x} = 2 y \) with \( y = e^{2x} \), where differentiation leads to a result directly proportional to the original function.
  • Also, the characteristic of exponential functions aids in solving both ordinary and partial differential equations, especially those representing financial models and natural phenomena.
Trigonometric Functions
Trigonometric functions, like sine and cosine, are inextricably linked with differential equations, especially ones dealing with periodic phenomena such as waves and oscillations. These functions provide solutions primarily because their derivatives are cyclic or periodic.
When a differential equation has solutions of the form \( y = \sin(x) \) or \( y = \cos(x) \), it often implies the equation involves rotating or oscillatory motion.
  • The equation \( \frac{d^2 y}{d x^2} = -4 y \) demonstrates this, with solutions involving sine and cosine functions due to their natural cycles.
  • Trigonometric function solutions are essential in physics and engineering for describing and predicting wave patterns, harmonic oscillators, and signal analysis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. If \(f(1)=5,\) then (1,5) could be a critical point of \(f\)

Give an example of: A logistic differential equation for a quantity \(P\) such that the maximum rate of change of \(P\) occurs when \(P=75\)

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(g(0)=1\) and \(g(x)\) is increasing for \(x \geq 0,\) then \(f(x)\) is also increasing for \(x \geq 0.\)

Policy makers are interested in modeling the spread of information through a population. For example, agricultural ministries use models to understand the spread of technical innovations or new seed types through their countries. Two models, based on how the information is spread, follow. Assume the population is of a constant size \(M\) (a) If the information is spread by mass media (TV, radio, newspapers), the rate at which information is spread is believed to be proportional to the number of people not having the information at that time. Write a differential equation for the number of people having the information by time \(t .\) Sketch a solution assuming that no one (except the mass media) has the information initially. (b) If the information is spread by word of mouth, the rate of spread of information is believed to be proportional to the product of the number of people who know and the number who don't. Write a differential equation for the number of people having the information by time \(t .\) Sketch the solution for the cases in which (i) No one \(\quad\) (ii) \(5 \%\) of the population (iii) \(75 \%\) of the population knows initially. In each case, when is the information spreading fastest?

Since \(1980,\) textbook prices have increased at \(6.7 \%\) per year while inflation has been \(3.3 \%\) per year. \(^{7}\) Assume both rates are continuous growth rates and let time, \(t,\) be in years since the start of \(1980 .\) (a) Write a differential equation satisfied by \(B(t),\) the price of a textbook at time \(t\) (b) Write a differential equation satisfied by \(P(t),\) the price at time \(t\) of an item growing at the inflation rate. (c) Solve both differential equations. (d) What is the doubling time of the price of a textbook? (e) What is the doubling time of the price of an item growing according to the inflation rate? (f) How is the ratio of the doubling times related to the ratio of the growth rates? Justify your answer.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.