Variation of parameters and the constraint

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Discussion Overview

The discussion revolves around the method of variation of parameters in solving second-order linear differential equations, specifically focusing on the use of constraints when substituting arbitrary constants with functions. Participants explore the justification for introducing a constraint on the functions involved and its implications for the solution space.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the justification for the constraint u'y1 + v'y2 = 0, noting that it simplifies the problem but questioning its validity.
  • Another participant suggests that the constraint allows u and v to behave partially like constants, which is a key aspect of the original solution for the homogeneous part of the ODE.
  • A different participant asks whether constants can be treated as functions of x, seeking clarity on the justification for treating u and v as constants in a limited sense.
  • One participant emphasizes that the constraint is a condition imposed to facilitate solving the differential equation, implying that without it, finding the unknown functions would be challenging.
  • A later reply raises a question about whether imposing this constraint discounts other possible solutions that may exist, indicating a concern about the completeness of the solution set.

Areas of Agreement / Disagreement

Participants generally agree on the necessity of the constraint for solving the differential equation, but there is uncertainty regarding the implications of imposing such a constraint and whether it excludes other potential solutions.

Contextual Notes

There is an acknowledgment of the limitations of the approach, particularly regarding the treatment of u and v as constants and the potential exclusion of other solutions due to the imposed constraint.

randybryan
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I have already read one thread on Lagrange's method of variation of parameters and it was very useful, but I am still confused about the use of the constraint.

If the solution to the homogeneous second order equation contains two functions, with arbitrary constants:

y= Ay1 + By2

Lagrange replaces constants by unknown functions so the solution becomes

y=u(x)y1 + v(x)y2

Then my notes say that inserting this solution into the differential equation provides a relationship between u, v and f(x). (Where the 2nd order linear diff equation is a0y'' +a1y' +a0y = f(x) )

But since there are two functions we need a second relation (i.e. we want a single solution, not a family of solutions). Therefore we are free to provide a constraint on the form of u and v (provided it leads to a solution)

Then my notes say, observe that adopting the relation:

u'y1 + v'y2 = 0

will make things easier.

This is where I'm confused. I mean, I can see that it trivially makes things easier by simply removing half of the equation to solve, but how can it be justified? And why not make y1'u + y2'v = 0? Mentally I equate it to someone building a house, but leaving out the electrical circuitry because it would make the building process easier.

I should point out I'm tired and 'mathed' out. Got to that point where things that made sense before suddenly don't make sense.
 
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Think of it as partially trying to make u and v behave as if they were constants, which was the original situation, when you obtained the general solution for the homogeneous part of your linear ODE; think about it, if they were indeed constant, u' would be 0, and so would be v'.
 
Ok thanks, I can see the logic behind that. Can constants still be treated as functions of x? Otherwise how can you justify momentarily treating u and v as constants?
 
Be careful! Notice the part where I say "partially". They're not actual constants, they are functions of x, but just forced to partially behave like constants with that additional equation that is introduced.
 
The way I understand this concept is, it is a condition we are FORCING to happen in order to be able to solve the differential equation. If we did not force this condition, it would be impossible to solve for the two unknown functions.
 
Ok thanks again, but one last question. By 'forcing' this condition are we discounting other possible solutions that may exist? I'm not saying we can find them without the constraint, but could there be other solutions?
 
Thats a good question. I will have to do some research to find out. :)
 

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