/*! 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 385 Solve the equation: \(\left(d^{2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 17: Problem 385

Solve the equation: \(\left(d^{2} y / d x^{2}\right)=-y\)

Short Answer

Expert verified
The general solution for the given differential equation \(\left(d^{2} y / d x^{2}\right)=-y\) is \(y(x) = A\sin(x) + B\cos(x)\), where A and B are constants that can be determined based on the initial conditions of the problem.

Step by step solution

01

Guess a solution

We will guess the solution of the given equation takes the form of: \(y(x) = A\sin(kx) + B\cos(kx)\) where A, B, and k are constants that we will determine.
02

Find the first and second derivatives of y(x)

Calculate the first derivative of y(x) with respect to x: \(\frac{dy}{dx} = Ak\cos(kx) - Bk\sin(kx)\) Now, calculate the second derivative of y(x) with respect to x: \(\frac{d^2y}{dx^2} = -Ak^2\sin(kx) - Bk^2\cos(kx)\)
03

Substitute the second derivative into the equation and solve for k

Substitute the second derivative into the given differential equation: \(-Ak^2\sin(kx) - Bk^2\cos(kx) = -y\) Using our solution for \(y(x) = A\sin(kx) + B\cos(kx):\) \(-Ak^2\sin(kx) - Bk^2\cos(kx) = -A\sin(kx) - B\cos(kx)\) We see that this equation is true if \(k^2 = 1\), since sine and cosine functions are linearly independent. This gives us two possible values for k: \(k = 1\) or \(k = -1\)
04

Write the general solution for the differential equation

Using the values for k, we can write the general solution for the given differential equation: \(y(x) = A\sin(x) + B\cos(x)\) where A and B are constants that can be determined based on the initial conditions of the problem.

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.

Differential Equation
Differential equations are mathematical equations that relate some function with its derivatives. In a physical context, these functions often represent physical quantities while the derivatives represent rates of change. The order of a differential equation refers to the highest derivative it contains. The equation in the exercise, \(\frac{d^{2} y}{dx^{2}} = -y\), is a second-order linear differential equation because the highest derivative is the second derivative and it can be rewritten in the form \(y'' + py' + qy = 0\) where \(p\) and \(q\) are coefficients, which in this case are 0 and -1 respectively.
Second Derivative
The second derivative of a function is the derivative of the derivative of that function. In the given exercise, the second derivative \(\frac{d^{2} y}{dx^{2}}\) represents the acceleration of a system if \(y\) were to represent position with respect to time. This concept is essential in physics, as it relates to how the velocity of an object changes over time, which is itself the time rate of change of position. By calculating the second derivative of \(y(x)\), we determine how the rate of change of \(y\) itself changes.
Harmonic Oscillator
The harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. The differential equation \(\frac{d^{2} y}{dx^{2}} = -y\) describes such a system. The solutions to this equation are sinusoidal functions, as demonstrated in the exercise, which is characteristic of simple harmonic motion.
Linear Independence
Two functions are said to be linearly independent if no constant multiple of one is equal to the other. In the context of differential equations, this is important because it ensures a diverse set of solutions. The sine and cosine functions used in the exercise solution are classic examples of linearly independent functions. We can also use the Wronskian determinant to formally prove their independence, solidifying the foundation for the general solution involving both \(A\) and \(B\).
Initial Conditions
Initial conditions specify the state of a system at the beginning of a process. When solving differential equations, these conditions allow us to find the specific solution that corresponds to a particular scenario. For example, in the exercise, if we were given the initial position and velocity of the oscillator, we could use those conditions to solve for the constants \(A\) and \(B\), determining the particular solution that matches the physical circumstances.

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

Solve \(p^{2}-4 y=0\) where \(p=(d y / d x)\)

Solve the differential equation: $$ y\left(d^{2} y / d x^{2}\right)+(d y / d x)^{2}=1 $$

Solve the differential equation: $$ \left(d^{2} y / d x^{2}\right)=\cos 2 x $$

Solve \(y^{\prime \prime}+k^{2} y=0\) by reduction of order.

Solve \(x y^{\prime \prime}-y^{\prime}=3 x^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.