Question

Hello. I just want to ask a question.
In the Methods of Undetermined Coefficients, what if you can't even factor the equation given?

Ex: (D^2 +4)y = 12cosx
How can you answer this? o.o

Answer 1

Does \(D^2y\) mean the second derivative of \(y\)? I've seen this notation before, but have never worked with it.

Answer 2

If the equation is what I think it is: \(y''+4y=12\cos x\), you would make a guess based on the derivatives present on the left side.

If you let \(y_p=A\cos x+B\sin x\), then you have \(y_p''=-A\cos x-B\sin x\). Since there are no sines on the right side, you can simply let \(y_p=A\cos x\), so that \(y_p''=-A\cos x\).

So, substituting into the original equation,
\[-A\cos x+4A\cos x=12\cos x\\
3A=12\\
A=4\]
So the nonhomogeneous solution is \(y_p=4\cos x\).

Answer 3

well, yes. it is the 2nd derivative of y.

Answer 4

Now I'll just need the auxillary equation..