Answer to Question #200300 in Discrete Mathematics for melaku

Question #200300
 Solve the recurrence relation of  an+4 + 2an+2 + an=0, n greater than or equal to zero .
1
Expert's answer
2021-06-01T16:08:30-0400

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.


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