Solve the recurrence relation of an+4 + 2an+2 + an=0, n greater than or equal to zero .

The Answer to the Question
is below this banner.

Can't find a solution anywhere?

NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

Solution. The characteristic equation has the form

$\lambda^4+2\lambda^2+1=0$

Let y=λ2. As a result, we get the quadratic equation

$y^2+2y+1=0$

$(y+1)^2=0$

The solution to this equation

$y=-1$

Back to substitution

$\lambda^2=-1$

As result

$\lambda_1=-i$

and

$\lambda_2=i$

The solution to the equation can be represented as

$a_n=(C_1+nC_2)(-i)^n+(C_3+nC_4)i^n$

where C1, C2, C3 and C4 are constants.

Answer.

$a_n=(C_1+nC_2)(-i)^n+(C_3+nC_4)i^n$

where C1, C2, C3 and C4 are constants.