- #1

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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.