Thread: Second-Order Linear Non Homogeneous ODE with Constant Coefficients

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:

Originally Posted by yianni

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:

Why is the trial particular solution not:



CB

Originally Posted by yianni

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:

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:

Ah, I see my mistake.

Thanks guys; solved.