This note contains some theorems that refer to the existence and uniqueness of the solution to the ODE.
Theorem 1. Consider the nth-order linear differential equation y(n)+p1(t)y(n−1)+p2(t)y(n−2)+…+pn(t)=f(t). If all coefficients p1(t), p2(t), ..., pn(t) and f(t) are continuous on the interval (a,b), the equation has the unique solution which satisfies the given initial conditions y(t0)=y0, y′(t0)=y0′, ..., y(n−1)(t0)=y0(n−1), where t0 belongs to the interval (a,b).