How Do You Solve a Second Order Linear Differential Equation?

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Homework Help Overview

The discussion revolves around solving a second order linear differential equation represented as -(ck^2)U(k,t)=d^2/dt U(k,t). Participants are exploring the general solution for this equation, which is related to simple harmonic motion (SHM).

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to rearrange the equation into a standard form and seeks guidance on the form of the solution. Some participants suggest potential solutions, while others question the completeness of these solutions and the role of integration constants.

Discussion Status

The discussion is active, with participants offering differing views on the solution's form. There is an acknowledgment of the need for arbitrary functions in the general solution, and some participants are correcting earlier statements regarding the constants involved.

Contextual Notes

Participants are navigating assumptions about the nature of the solution and the implications of constants that may depend on k. There is a focus on ensuring that the solution reflects the characteristics of the differential equation accurately.

captainjack2000
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Homework Statement


I have the equation -(ck^2)U(k,t)=d^2/dt U(k,t). And I need to find the general solution.


Homework Equations





The Attempt at a Solution


I can rearrange this into the form d^2 U/dt^2 + (ck^2)U(k,t)=0 but I am not sure of the form of the solution to this equation.

could someone please give me pointer in the right direction
 
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Is it just U(k,t) = Acos(ckt)+Bcos(ckt)?
 
Yeah, it's just the differential equation for SHM so that is the solution.
 
No it's not. You will get a constant when you do the integration with respect to t, and this constant can depend on k. So the general solution is what you wrote + f(k), where f(k) is an arbitrary function of k.

EDIT: You should have (√c)k instead of ck in the sin and cos.
 
Oops! I'm extremely sorry, my previous post is wrong. The constants A and B are the constants of integration, and they should be arbitrary functions of k.
 

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