existence and uniqueness of solutions for nonlinear 2nd order ode with Cauchy boundary conditions

Question

What are the known results on the existence and uniqueness of solutions for nonlinear second order ordinary differential equations with Cauchy boundary conditions?

Answer 1

You can always "unroll" an ODE involving higher-order derivatives into a system involving only first-order derivatives. After having done so, it is possible to apply the Picard-Lindelöf theorem for existence and uniqueness (see also the Peano existence theorem for existence without uniqueness).

For example, consider the nonlinear second order ODE $$ y^{\prime\prime}(t)=f(t,y(t),y^{\prime}(t)). $$ Let $z(t)=y^{\prime}(t)$ so that $z^{\prime}(t)=y^{\prime\prime}(t)$. Then, the ODE can be rewritten as the system \begin{align*} y^{\prime}(t) & =z(t);\\ z^{\prime}(t) & =f(t,y(t),z(t)). \end{align*} Letting $x(t)=(y(t),z(t))$, we can put this into the familiar form $$ x(t)=f(t,x(t)) $$ and apply Picard-Lindelöf (or Peano).