Differentials:Please to if my work is correct

I just want to see if my work is correct.

This problems comes in for parts. I am concerned about the last three. I think the processes should be longer but I don't see how I can make them longer.

4.8 #1

y`` + y = 0

a.

e^xsolve with: e^x

e^x y`` + y = 0

a y"+b y' +c y=0

a y’’ +b y’ + c =0

a e^(y’’ x) + b e^(y’ x)

e^(y)(a cos(x) +b sin(x))

This is not a solution

b.Sin x

sin x y'' + y = 0

1/2 i e^-i^x y``(x)- 1/2 i e^i^x y``(x)+y(x)=0

y(x)+sin(x) = cos(x) y''(x)

Solution

c.Cos x

cos x y'' + y = 0

1/2 e^(-i x) y''(x)+1/2 e^(i x) y''(x)+y(x) = 0cos(x) y''(x)+y(x) = 0Solution

d.Sin x – cos x

-1/2 e^(-i x) y''(x)-1/2 e^(i x) y''(x)+y(x)+1/2 i e^(-i x)-1/2 i e^(i x) = 0

Solution