A restriction within Variation of Parameters

[Sparky_]

Within the description for the variation of parameters procedure is the restriction:
y1u1' + y2u2' = 0.

Can you explain this restriction, it is not obvious to me, I do not have an explanation where this comes from.

Is it related to
$$u[ \frac {dy}{dx} + P(x)y] = 0$$

from solving first order equations?

Thanks
Sparky_

 

[HallsofIvy]

First the whole point of "variation of parameter" is that you are looking for a solution of the form y(x)= u1y1+ u2y2, where y1 and y2 are solutions to the related homogeneous equation. The important thing to remember is that there are an infinite number of such solutions. In fact, given any solution y(x), you could make up an infinite number of pairs, u1, u2, that give y(x).

The point is that the restriction u1'y1+ u2'y2= 0 only restricts which of those infinite number of possible u1, u2 we are looking for. Of course, the reason for restricting in that way is that it prevents second derivatives from showing up in the final equations.