Image. please help!!

Step-by-step explanation:

not sure sorry

(1)

Therefore the complete solution is

(2)

Therefore the complete solution is

Step-by-step explanation:

Given differential equation is

y''+4y = g(x) is associated  with y''+4y=0.

The auxiliary equation is

m²+4=0

⇒m²= -4

The complimentary function is

Case 1:

g(x)= sinx

Then y''+4 y= sin x

F(D)≡ (D²+1)     [Here ]

Therefore

        [ replace D² by -1² and F(D)≠0]

The complete solution is

Differentiating with respect to x

The initial condition are y(0) =1 and y'(0)=4

and

Therefore the complete solution is

Case 2:

g(x)=0

Then

The complete solution is

Differentiating with respect to x

The initial condition are y(0) =1 and y'(0)=4

and

⇒B=2

Therefore the complete solution is