[[Second Order Linear Scalar Ordinary Differential Equation]] # Properties **Existence and uniqueness of solution to IVP**: Given a function $f(x_{2},x_{1},t)$ such that $f,\frac{ \partial f }{ \partial x_{1} },\frac{ \partial f }{ \partial x_{2} }$ are continuous functions for $a_{1}<x_{1}<a_{2},\; b_{1}<x_{2},b_{2},$ and $t_{1}<t<t_{2}.$ [[Picard–Lindelöf theorem|Picard–Lindelöf theorem]] asserts that there exists a unique solution of initial value problem of $x''(t)=f(x'(t),x(t),t)$ subject to $x(t_{0})=x_{0}$ and $x'(t)=v_{0}$ where $a_{1}<x_{0}<a_{2},\; b_{1}<v_{0}<b_{2}$ and $t_{1}<t_{0}<t_{2}.$