Here it is:
So the auxiliary/characteristic equation will look like:
Solving that gives:
And so we'll get the following general homogeneous solution:
So now I try to find the particular non homogeneous solution, trying:
Finding the first and second derivative of y particular:
Then, when I try to compile all of this, to then equate coefficients and solve for alpha and beta, I realise that the x terms do not cancel out (as they should) and I cannot go any further (I don't think).
It will eventually look like this:
Have I made an error somewhere?
Any help is appreciated.
NB: It may/may not help, but in the previous part of the question we've already proved that when:
I suspect you put that "x" multiplying the sine and cosine in because you saw "(A cos(4x)+ Bsin(4x))" in the homogeneous solution. However, you really have and which are NOT equivalent to "cos(4x)" and "sin(4x)". More simply, the solutions to you characteristic equation are 3+ 4i and 3- 4i while just sin(4x) and cos(4x) correspond to 4i and -4i- quite different solutions.
Finding the first and second derivative of y particular:
Then, when I try to compile all of this, to then equate coefficients and solve for alpha and beta, I realise that the x terms do not cancel out (as they should) and I cannot go any further (I don't think).
It will eventually look like this:
Have I made an error somewhere?
Any help is appreciated.
NB: It may/may not help, but in the previous part of the question we've already proved that when: