Energy in SHM

We found before that the potential energy stored in a spring is  and the kinetic energy is . The conservation of mechanical energy says that

where the total energy is

That is

Thus E is constant. However for a spring x and v are always changing. Can we be sure that E is always constant ?

Example For SHM, show that the total energy is always constant even though K and U always change.

Solution Recall that for SHM we have x = x_m cos t and v = -xm sin t. Thus

and

Thus U and K always change. Let's add them.

but we previously found that  giving

which is always constant because the amplitude x_m is constant!