KILLER 2nd ODE (inhomogeneous) XD

[Zomboy]

Ok, here goes:

Homework Statement

So I've come across this 2nd ODE which I am to "solve ... for a general solution":

d^2y / dx^2 - dy/dx + y = cos(x) - sin(x)

and then evaluate the "particular solution" using the boundary conditions y=L when x=0 (also, dy/dx = 0)

The Attempt at a Solution

I can't type out the whole of my working because its really long and would be impossible to follow so I'll try and sum up what I've got:
1) found the general solution of the * equivalent* homogeneous equation... which came out with imaginary values. I then converted this into trigonometric form (as opposed to using imaginary exponentials) which is in the form of:

exp(1/2 x) ( C sin((sqrt(3)/2)x) + D cos("") )
2) I then guessed at the particular solution which I'm thinking looks like:

(a-b)( cos(x) - sin (x) )
3) Added ^^these^^ together to get the "General Solution" (y=...) of the original equation. Which looks something like (but with more coefficients and stuff:

e^... (cos + sin) + ( cos - sin ) {you get the idea} 4) Trying to evaluate this however lead me to some nasty unsolvable simultaneous equations...

any advice? can you spot my mistake? do I actually just need to solve "2)" to get the answer, I'm confused now...

this is driving me absolutely crazy.

 

[Dickfore]

The particular solution should be of the form:
$$Y_p(x) = A \, \cos x + B \, \sin x$$

Determine A and B by plugging this in the ODE. Tell us what you get.

 

[Ray Vickson]

Ok, here goes:

Homework Statement

So I've come across this 2nd ODE which I am to "solve ... for a general solution":

d^2y / dx^2 - dy/dx + y = cos(x) - sin(x)

and then evaluate the "particular solution" using the boundary conditions y=L when x=0 (also, dy/dx = 0)

The Attempt at a Solution

I can't type out the whole of my working because its really long and would be impossible to follow so I'll try and sum up what I've got:



1) found the general solution of the * equivalent* homogeneous equation... which came out with imaginary values. I then converted this into trigonometric form (as opposed to using imaginary exponentials) which is in the form of:

exp(1/2 x) ( C sin((sqrt(3)/2)x) + D cos("") )



2) I then guessed at the particular solution which I'm thinking looks like:

(a-b)( cos(x) - sin (x) )



3) Added ^^these^^ together to get the "General Solution" (y=...) of the original equation. Which looks something like (but with more coefficients and stuff:

e^... (cos + sin) + ( cos - sin ) {you get the idea}





4) Trying to evaluate this however lead me to some nasty unsolvable simultaneous equations...

any advice? can you spot my mistake? do I actually just need to solve "2)" to get the answer, I'm confused now...

this is driving me absolutely crazy.

Suggestions:
(1) Use a Green's function method http://www.math.umn.edu/~olver/pd_/gf.pdf ; or
(2) Use Variation of Parameters http://en.wikipedia.org/wiki/Variation_of_parameters

Both methods are standard, but (2) is probably better known.

RGV