- #1
samoth2
- 2
- 0
Homework Statement
Find the general solution
Homework Equations
x(t+2)-3x(t+1)+2x(t)=3*5^t+sin(0.5πt)
The Attempt at a Solution
I start out by solving the homogeneous equation and end up with the two roots 1 and 2.
Then I try to use the method of undetermined coefficients to find a particular solution.
I guess that the solution is of the following form (I should probably include a cos term as well)
u(t)=C*3*5^t+D*sin(0.5πt)
then
u(t+1)=C*3*5^(t+1)+D*sin(0.5π(t+1))
and
u(t+2)=C*3*5^(t+2)+D*sin(0.5π(t+2))
I then insert these equations into the original equation to get
C*3*5^(t+2)+D*sin(0.5π(t+2))-3*(C*3*5^(t+1)+D*sin(0.5π(t+1)))+2*(C*3*5^t+D*sin(0.5πt))=3*5^t+sin(0.5πt)
move the terms around a bit to get
3*5^t(C*5^2-3*C*5+2*C)+D*sin(0.5π(t+2)-3*D*sin(0.5π(t+1))+2*D*sin(0.5πt)=3*5^5+sin(0.5πt)
From the first part I see that
(C*5^2-3*C*5+2*C)=1
so
C=1/12
I would like to do something similar for the sin part of the expression, but I'm not sure how to handle it.
I tried with sin(0.5π(t+2))=sin(0.5πt+π)=-sin(0.5πt) but then I don't know what to do with sin(0.5(t+1)).
If anyone has any hints, tips or tricks I would be happy.
I hope the equations are understandable otherwise I'll post them as a picture.
Thanks in advance.