- #1
randybryan
- 52
- 0
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.
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.