goonking said:
yes, this is much easier than the Yn and Yp stuff. I just need to connect the pieces to the puzzle.
This problem is a simplified version of the business with a complementary solution and a particular solution. It helps to think in terms of vectors.
This problem has two equations (I have replaced u and v by x and y, respectively, in hopes that it will make things easier to understand):
x - 2y = 0 -- the homogeneous equation
x - 2y = 3 -- the nonhomogeneous equation
The first equation has an infinite number of solutions -- vectors of the form C<1, 1/2>; i.e., any constant multiple of the vector <1, 1/2>. As long as the first coordinate is two times the second coordinate, the vector is a solution to the homogeneous equation. Or, to put in another way, the point (C, C/2) is on the line whose equation is x - 2y = 0, or equivalently, y = (1/2)x.
The second equation also has an infinite number of solutions. Among them are the vectors <3, 0>, <5, 1>, and so on (or the points (3, 0), (5, 1), and so on). You can pick anyone of these as a particular solution -- say, <3, 0>
To get to any point on the line x = 2y + 3, you can go from the origin to <3, 0> and then along the line in the proper direction. To get to, say, (7, 2), go to right right from the origin to (3, 0), and then go along the vector from (3, 0) to (7, 2) by the appropriate multiple of <1, 1/2>. In terms of vectors, <7, 2> = <3, 0> + 4<1, 1/2>.
To get to an arbitrary point on the line x = 2y + 3, it's possible to solve for the constant C that does the trick, so that <x, y> = <3, 0> + C<1, 1/2> for an appropriate value of C.
This all ties into differential equations, with this pair as an example:
y' - 2y = 0 (homogeneous)
y' - 2y = 3 (nonhomogeneous)
The solutions to the homogeneous equation (the complementary solutions) are y
c(t) = Ae
2t, where A is an arbitrary constant.
A solution to the nonhomogeneous equation (a particular solution) is y
p(t) = -3/2, which I got by an educated guess.
The general solution to the nonhomogeneous equation is made up of the complementary solutions, plus the particular solution:
y(t) = y
p(t) + y
c(t) = -3/2 + Ae
2t
Because y
c(t) is a solution to the homogeneous equation, it must be true that y
c' - 2y
c = 0, so adding it to the general solution doesn't change the right side of the equation.
##\frac{d}{dt}(y_c(t) + y_p(t)) - 2(y_c(t) + y_p(t) = \frac{d}{dt}(Ae^{2t} - \frac 3 2) - 2(Ae^{2t} - \frac 3 2} = 2Ae^{2t} - 0 - 2Ae^{2t} + 3 = 3##
Thinking about this in terms of vectors, the particular solution gets us up to the space (of functions) that is generated by ##Ae^{2t}##, and the constant A gets us to a particular function in that space.
