Question

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

Answer 1

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.