First order differential equation transforms

[cue928]

I am to find a general solution of the system of problems below. I have done so for x(t) but am unsure how to find it for y(t)...
x'=y, y'=-x
x''=y=-x
x''+x=0
Characteristic eq: r^2 + 1= 0, r = +/- i
x(t) = A cos(t) + B sin(t)
How do I go about calculating y(t), which th book shows as being y(t)=B cos(t)+Asin(t)

Similarly, I calculated x(t)=A cos (2t) + B sin(2t), but am unsure how to get the book's version of y(t)=4B cos 2t - 4A sin(2t)

 

[Char. Limit]

you have x'=y, don't you? Could that help?

 

[cue928]

I apologize, I looked at y(t) wrong; y(t) is actually B cos(t)-A sin(t).

 

[HallsofIvy]

Yes, x(t)= Acos(t)+ Bsin(t). Since the first equation is x'= y, as Char. Limit said, that leads to y= x'= -A sin(t)+ B cos(t) which is what you give in your second post.

As for showing it is the same as y(t)=4B cos 2t - 4A sin(2t), you can't- it simply is not true. Are you sure you are not looking at the wrong problem?