I have an equation y''+y=0 with y(0)=1 and y'(0)=0

It has general solution of y=Asin(x)+Bcos(x) ?

with condition gives 1 = 0 + B so B=1

If so y' = Acos(x)-sin(x)

with condition gives 0 = A - 0 so A=0

Therefore the solution is y = cos(x)

Am I going about this the write way?

And if so does anything change to the general solution if I am solving

y''+4y=0 with y(0)=1 and y'(0)=0 ?

Thank you!