Can an Exponential Factor Create a New Solution for a Second Order ODE?

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Discussion Overview

The discussion centers on the possibility of creating a new solution for a second-order ordinary differential equation (ODE) by multiplying existing solutions by an exponential factor. The scope includes theoretical considerations and the justification of mathematical procedures related to differential equations.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether multiplying a solution of a second-order ODE by an exponential factor can yield a new solution.
  • Another participant clarifies that in a second-order linear differential equation, if one solution is known, the general solution can be formed by multiplying it by a function of the form f(x)u(x).
  • A different participant presents a specific case where two particular solutions diverge at infinity but suggests that multiplying them by exp(-z^2 / 4) leads to better behavior, although they struggle to justify this mathematically.
  • One participant asserts that generally, multiplying a solution by another function does not produce a new solution to the differential equation.

Areas of Agreement / Disagreement

Participants express differing views on the validity of creating new solutions through multiplication by an exponential factor, indicating that the discussion remains unresolved.

Contextual Notes

There are unresolved mathematical steps regarding the justification of the proposed multiplication of solutions and its implications for the differential equation.

intervoxel
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Is it possible to form a new solution of a second order ODE by multiplying it by an exponential factor?
 
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Not clear what you mean. In a second-order LINEAR diff eq, if you have one solution u(x) you can find the general solution by trying the form f(x)u(x) ... This should be in standard ODE textbooks.
 
What I mean is that I have two particular solutions even and odd that diverge at infinity, but I noted that if I multiply them by exp(-z^2 / 4) they behave properly. I'm trying to justify this procedure. Substituting the product of each back into the differential equation doesn't seem to work.
 
In general, no. Multiplying a solution to a d.e. by another function, exponential or not, does NOT give a new solution.
 

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