What are the known results on the existence and uniqueness of solutions for nonlinear second order ordinary differential equations with Cauchy boundary conditions?
[Math] existence and uniqueness of solutions for nonlinear 2nd order ode with Cauchy boundary conditions
nonlinear systemordinary differential equations
Best Answer
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).