There's a further way of seeing the solution for a harmonic oscillator. We can use complex numbers. This involves the use . An imaginary number is a multiple of , like , , etc. A complex number is the sum of a regular (real) number and an imaginary number. But actually these complex numbers aren't so complex and have a lot of really nifty properties.
The use of complex numbers is extremely important in physics and is used all the time in a wide variety of fields, so this should give you an introduction as to how to use them and what they mean.
Let's guess another functional form and see if it solves eqn. 1.1
This time guess
So let's calculate the velocity
Notice the negative sign! This says that
So we get that is imaginary.
We therefore know that one solution to the harmonic oscillator problem is
It turns out that another solution to this problem would be to sum these two solutions
together. So the general solution to this problem can be written as the sum of
the two above solutions. If you don't want to keep writing all these pesky constants,
you could say that the solution to this equation was
Similarly, we learned previously that the general solution can also be written as
They both solve the same equation 1.1 which can be rewritten