This is a problem from the book:
$$x_1' = x_2\\ x_2' = -x_1\\ x_1(0) = 2\\ x_0(0) = 0$$
The problem says transform the system of the 1st order differential equations into a single differential equation and solve it.
My approach:
I get $x_1'' + x_1 = 0$ by eliminating $x_2$.
How do I use the initial condition?
How to finish the problem?
Thanks!
You already have the equation $x_1'' = - x_1$, as you rightly pointed out. You also have $x_1(0) = 2$ and $x_1'(0) = x_2(0) = 0$. This means you have a standard 2nd degree linear differential equation problem:
$$x_1'' + x_1 = 0\\ x_1(0) = 2\\ x_1'(0) = 0$$
which you can solve using standard technicques for LDE-s