# ODE (solve for particular integral) am i right?

1. Feb 24, 2009

### soonsoon88

ODE (solve for particular integral) am i right??

1. The problem statement, all variables and given/known data
That is all the examples that i know to determine the particular solution.
are all of them correct? Note: e= exponent

a)x^3 ==> Ax^3 + Bx^2 + Cx + D
b)xe^2 ==> e^2(Ax +B)
c)sin x ==> Asin x + Bcos x
d)cos x ==> Asin x + Bcos x
e)sin x + cos x ==> Asin x + Bcos x + Csin x + Dcos x
f)sin x + xcos x ==> Asin x + Bcos x + (Cx+D)sin x + (Ex+F)cos x
g)x^2(sin x) ==> (Ax+B)sin x + (Cx+D)cos x
h)e^x (sin x) ==> e^x( Asin x + Bcos x)
2. Relevant equations
Am i right?
is it still got another type of particular?
that is all i know only.

3. The attempt at a solution

2. Feb 24, 2009

### jimmypoopins

Re: ODE (solve for particular integral) am i right??

that looks about right, except for e) and g). also, on b, it it supposed to be xe^(2x)?

Asin x + Bcos x + Csin x + Dcos x = sinx(A+C) + cosx(B+D). since A, B, C, D are just constants, you can tell that you don't realy need the C or the D.

on g), you're missing an x^2 term.

3. Feb 24, 2009

### HallsofIvy

Staff Emeritus
Re: ODE (solve for particular integral) am i right??

One caveat: If a function of that form is already a solution to the associated homogeneous equation, then you will need to multiply by 4.

For example, if the problem is y"- 5y'+ 6y= ex, so the characteristic equation is r2- 5r+ 6= (r- 3)(r- 2)= 0, which has roots 2 and 3, so that the solutions to the homogeneous equation are e2x and e3x, then, yes, I would try a particular solution of the form y= Aex.

But if the problem is y"- 4y'+ 3y= ex, so the characteristic equationj is r2- 4r+ 3= (r-1)(r- 3)= 0, which has roots 1 and 3, so that the solutions to the homogeneous equation are ex and e3x, then Aex will give 0, not ex, for any A. Now I would try y= Axex.

If the problem is y"- 2y'+ y= ex, so that the characteristic equation is r2- 2r+ 1= (r-1)2= 0, which has 1 as double root, so that the solutions to the homogeneous equation are ex and xex, then I would have to tr Ax2ex.