What are the steps for solving a 2nd order differential equation?

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Homework Help Overview

The discussion revolves around solving a second-order differential equation, specifically in the context of non-homogeneous linear ordinary differential equations (ODEs). Participants explore the relationship between solutions of different forms of the equation.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the implications of scaling a solution by a constant and the structure of the general solution, including the complementary function and particular integral.

Discussion Status

Some participants confirm the correctness of the proposed relationships between solutions, while others express uncertainty about the straightforwardness of the problem. There is a mix of exploration and validation of ideas without a clear consensus.

Contextual Notes

The original poster indicates difficulty with the problem, suggesting that it may not be a straightforward calculation. The discussion includes references to specific forms of the differential equation and their solutions.

wombat4000
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Firstly, suppose y1 is a solution to y''+ay'+by=g(x). What then is a solution of y''+ay'+by=kg(x), where k is an arbitrary non-zero constant?

Secondly, note that a general solution to a non-homogenous linear ODE is its homogenous solution plus one particular solution to its nonhomogenous ODE.
 
solution to y''+ay'+by=kg(x) is ky1?

so the solution to y''+ay'+by=3g(x) is the complimentyery function + particular integral = 3y1 + y2?
 
That should be correct.
 
thanks!
http://img237.imageshack.us/img237/3676/167uqnbsi6.gif
 
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