variation of parameters

the undetermined coefficient method may not prove out to be useful all the time ..we may need to follow a little different path...

equation we have is Y''+const.Y'+Y = G(x)

assume particular solution is Yp = u1y1+u2y2 ............(1)

Yp' = u1'y1+u1y1'+u2'y2+u2y2'

assume u1'y1 + u2'y2' = 0 then Yp' = u1y1'+u2y2'

Yp'' = u1'y1'+u2'y2'+u1y1''+u2y2''

putting all these values in (1)...we have

u1'y1'+u2'y2'+u1y1''+u2y2''+const. * u1y1'+const.u2y2'+u1y1+u2y2 = G(x)

u1(y1''+const.y1'+y1)+u2(y2''+const.y2'+y2)+u1'y1'+u2'y2' = G(x)

y1''+const.y1'+y1 = 0 as per the general solution ..we are left with

u1'y1'+u2'y2' = G(x) ........(2)

u1'y1+u2'y2 = 0 ...(3)

u1' = -u2'y2/y1

u2' = G(x)*y1 / (y2'y1-y2y1')..............(4)

u1' = -G(x)*y2 / (y2'y1-y2y1').............(5).

substituting 4 and 5 in (1)...we have

Yp = y1*integration(-G(x)*y2 / (y2'y1-y2y1'))+y2*integration(G(x)*y1 / (y2'y1-y2y1'))