8. For what value of k, if any, will y=ke?' +4cos(3x)be a solution to the differential equation y"+9y = 26e 22

(A) 2

(B)

13

5

(C) 26

(D) There is no such value of k.

k = 2

Step-by-step explanation:

As given,

the differential equation is - y'' + 9y = 26      .......(1)

Let the solution of the differential equation be y =

The auxiliary equation becomes

m² + 9 = 0

⇒m² = -9

⇒m = ±√-9

⇒m = ± 3i

So, the complimentary solution becomes

= A cos(3x) + B sin(3x)

Now,

Let the particular solution be = A

⇒ = -2A

and = 4A

Now,

equation (1) becomes

4A + 9 ( A )= 26

⇒4A + 9A  = 26

⇒13A  = 26

By comparing , we get

13A = 26

⇒A = = 2

∴ we get

= 2

so, the solution becomes

y = A cos(3x) + B sin(3x) + 2

As given , the solution y=k +4cos(3x)

So , by comparing with the solution , we get k = 2

So, the correct option is (A) 2.

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step-by-step explanation: