What guess should I use when the right hand side is a constant?

[CraigH]

What form should the particular solution of a 2nd order linear non homogeneous differential equation take when the right hand side is a constant?

if the differential differential equation has the form

Ay''+By'+Cy = g(x)

where g(x) is a constant

what form should the particular solution take?

I know if g(x) is an exponential the "trial" solution should be A(e^x)

If g(x) is trigonometric it should be A(sin(x))+B(cos(x))

And If it is a polynomial of degree n it should be A(x^n) + B(x^(n-1)) ... +C(x^0)

But what if the differential equation is just Ay''+By'+Cy = D

Should the trial solution be a polynomial of degree 0?

So I would try
y = D
y' = 0
y'' = 0

which would leave me with

CD=D

which is useless

Please Help!

Thanks!

 

[mfb]

Well, $y=\frac{D}{C}$ is a solution to your equation. It is not the only one, so you can add all solutions to Ay''+By'+Cy = 0. You can use both the exponential and the trigonometric approach (with complex numbers, they are the same anyway) to find them.

 

[CraigH]

Ahhh okay, thank you for your answer :D