# Solution to 2nd order ODE using the D operator method with 2 trig terms on RHS

1. Aug 9, 2012

### mitch_1211

Hey,

I have the DE

y'' -2y' + 3y = xsin(x) + 2cosh(2x)

Using the D operator as D = $\frac{dy}{dx}$ this becomes

(D2 -2D +3)y = xsin(x) + 2cosh(2x)

so yp = $\frac{1}{p(D^2)}$ operating on xsin(x) + 2cosh(2x)

(i think)

So i know if this was say $\frac{1}{p(D^2)}$ operating on sin(x)
i.e (D2 -2D +3)y = sin(x)
then yp = 1/(D2 -2D +3) * sin(x)

and you would substitute D2 = -($\alpha$)2 and proceed to solve.(where alpha is the coefficient of x in the argument of sin, here it is 1)

My question is, I have a term on the RHS which is polynomial times trig; xsin(x) and a trig term which can be treated as an exponential; 2cosh(2x)

I know I can split this into
yp = 1/(D2 -2D +3) * xsin(x) + yp = 1/(D2 -2D +3) * 2cosh(2x)

But I'm unsure of the 'rules' to use here, i.e for a single trig term you swapped D2
for -($\alpha$)2
But what do you do for a poly times a trig and for the cosh function?

Any points would be much appreciated, if all else fails I will have to resort to the method of undetermined coefficients