Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Suppose that you have found a particular solution xp(t) of the inhomogeneous equation (5.48) for a driven damped oscillator, so that Dxp=f in the operator notation of (5.49). Suppose also that x(t) is any other solution, so that Dx=f. Prove that the difference xxp must satisfy the corresponding homogeneous equation, D(xxp)=0. This is an alternative proof that any solution x of the inhomogeneous equation can be written as the sum of your particular solution plus a homogeneous solution; that is, x=xp+xh.

Short Answer

Expert verified
The difference xxp satisfies the homogeneous equation, proving x=xp+xh.

Step by step solution

01

Define the Given Functions

We are given that xp(t) is a particular solution of the inhomogeneous equation Dxp=f, and x(t) is any solution of the same equation Dx=f. Thus, both satisfy the inhomogeneous differential equation described by operator notation.
02

Express the Difference Function

We are asked to consider the function y(t)=x(t)xp(t). We hypothesize that this function satisfies the homogeneous equation. The task is to prove that D(y)=0.
03

Substitute into the Operator

Substitute y(t)=x(t)xp(t) into the differential operator D: D(y)=D(xxp)=D(x)D(xp).
04

Apply Given Differential Equations

Since D(x)=f and D(xp)=f, substitute these into the equation from the previous step:D(y)=ff=0.
05

Conclusion of the Proof

Since D(y)=0, the function y(t) satisfies the homogeneous equation. Consequently, any solution x(t) of the inhomogeneous equation can be expressed as the sum:x(t)=xp(t)+xh(t),where xh(t) is a solution to the homogeneous equation D(xh)=0.

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 Vaia!

Key Concepts

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

Driven Damped Oscillator
A driven damped oscillator is a system where a damping force and an external driving force act on an oscillating object. This is common in many real-world scenarios, such as in car suspension systems or pendulums affected by air resistance and periodic pushes.

In physics terms, the damping force usually takes away energy from the system, causing the oscillating motion to gradually decrease over time. The driving force, on the other hand, adds energy to the system, which can sustain or even increase the amplitude of the oscillations.
  • Damping force typically has a negative impact on motion, slowing it down.
  • Driving force is often external and periodic, designed to influence the oscillator.
Understanding the balance of these forces is crucial in predicting how the system will behave over time, often modeled with differential equations to quantify the effect mathematically.
Inhomogeneous Equation
An inhomogeneous equation in the context of differential equations involves a function on the right-hand side. This makes the equation non-uniform across the domain of solutions.

For a driven damped oscillator described earlier, this function is the external driving force, a term added to the differential equation of the system, represented as follows: Dx=f, where f is a non-zero function.
  • The presence of f introduces complexity, as solutions can't simply return to equilibrium without external influence.
  • Inhomogeneous equations require different techniques to solve, including finding a particular solution that satisfies the equation.
Repeatedly encountering such equations suggests systems affected by external factors, including electrical circuits, mechanical systems, and thermal processes.
Homogeneous Equation
A homogeneous equation is a special type of differential equation where the right-hand side is zero. This signifies that there are no external forces acting on the system.

In the context of our exercise, it is written as:D(xxp)=0. Understanding these allows us to focus solely on the intrinsic properties of the system itself.
  • Solutions to homogeneous equations only depend on initial conditions and natural tendencies of the system.
  • In mechanical terms, it means systems return to their base state without external disruption.
The beauty of homogeneous solutions often lies in their simplicity and purity, revealing fundamental characteristics of the oscillators or other systems described.
Particular Solution
The particular solution is a crucial component in solving inhomogeneous differential equations. It's a specific solution that satisfies the non-zero right side of the equation.

For the driven damped oscillator, it is the particular solution xp(t) that accommodates the external force f applied:
Dxp=f.
  • Finding this particular solution often involves making educated guesses or using known problem-solving methods like the undetermined coefficients or variation of parameters.
  • It represents the specific way the system responds to external influences, considering all forces applied.
Adding this to the general solution of the homogeneous equation gives us the overall response of the system.
Operator Notation
Operator notation is a powerful method of representing differential equations concisely. It employs mathematical operators to manage derivatives without expanding the full form each time.

In our scenario, the operator D is used to signify the differentiation process. We see it applied as:Dx=f for inhomogeneous equations, and D(xxp)=0 for homogeneous equations.
  • This method streamlines complex differential equations, making solutions more accessible and manageable.
  • It allows focusing on solving techniques instead of repeatedly dealing with the difference quotient.
Acknowledging notation differences can help in understanding the full spectrum of approaches needed for solving these equations efficiently.

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

An unusual pendulum is made by fixing a string to a horizontal cylinder of radius R, wrapping the string several times around the cylinder, and then tying a mass m to the loose end. In equilibrium the mass hangs a distance lo vertically below the edge of the cylinder. Find the potential energy if the pendulum has swung to an angle ϕ from the vertical. Show that for small angles, it can be written in the Hooke's law form U=12kϕ2. Comment on the value of k.

Write down the potential energy U(ϕ) of a simple pendulum (mass m, length l ) in terms of the angle ϕ between the pendulum and the vertical. (Choose the zero of U at the bottom.) Show that, for small angles, U has the Hooke's law form U(ϕ)=12kϕ2, in terms of the coordinate ϕ. What is k?

The potential energy of a one-dimensional mass m at a distance r from the origin is U(r)=U0(rR+λ2Rr) for 0<r<, with Uo,R, and λ all positive constants. Find the equilibrium position ro. Let x be the distance from equilibrium and show that, for small x, the PE has the form U= const +12kx2. What is the angular frequency of small oscillations?

The maximum displacement of a mass oscillating about its equilibrium position is 0.2m, and its maximum speed is 1.2m/s. What is the period τ of its oscillations?

Consider an underdamped oscillator (such as a mass on the end of a spring) that is released from rest at position xo at time t=0. (a) Find the position x(t) at later times in the form x(t)=eβt[B1cos(ω1t)+B2sin(ω1t)] That is, find B1 and B2 in terms of xo. (b) Now show that if you let β approach the critical value ωo your solution automatically yields the critical solution. (c) Using appropriate graphing software, plot the solution for 0t20, with xo=1,ωo=1, and β=0,0.02,0.1,0.3, and 1

See all solutions

Recommended explanations on Physics Textbooks

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.

Sign-up for free