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.

 

[tacman]

The restriction y1u1' + y2u2' = 0 is what is known as the "homogeneous condition" in the variation of parameters method. This condition is necessary in order for the method to work and produce a valid solution.

To understand this restriction, let's first review the variation of parameters method. This method is used to find a particular solution to a non-homogeneous linear differential equation of the form y'' + p(x)y' + q(x)y = g(x). In order to use this method, we must have the general solution to the corresponding homogeneous equation, which in this case is y'' + p(x)y' + q(x)y = 0. This general solution can be written as y = c1y1(x) + c2y2(x), where y1 and y2 are linearly independent solutions to the homogeneous equation and c1 and c2 are constants.

Now, the variation of parameters method involves finding two new functions, u1 and u2, that will be used to modify the constants c1 and c2 in the general solution. These functions are chosen such that when we substitute y = u1y1 + u2y2 into the non-homogeneous equation, the terms involving u1 and u2 will cancel out, leaving only g(x) on the right-hand side. This is where the restriction y1u1' + y2u2' = 0 comes into play. By setting this condition, we ensure that the terms involving u1 and u2 will indeed cancel out and we will be left with only g(x) on the right-hand side.

In summary, the restriction y1u1' + y2u2' = 0 is necessary in order for the variation of parameters method to produce a valid solution to the non-homogeneous equation. It ensures that the terms involving u1 and u2 will cancel out and we will be left with only the non-homogeneous term on the right-hand side. This restriction is related to the homogeneous condition u[ \frac {dy}{dx} + P(x)y] = 0, as both are necessary in order for the method to work and produce a valid solution.