4.2.2 Existence and uniqueness for non-linear second order ode

Now the ode is written in the form

\begin{align*} y^{\prime \prime } & =f\left ( x,y,y^{\prime }\right ) \\ y\left ( x_{0}\right ) & =y_{0}\\ y^{\prime }\left ( x_{0}\right ) & =y_{0}^{\prime }\end{align*}

Then if \(f\) is continuous at \(\left ( x_{0},y_{0},y_{0}^{\prime }\right ) \) and \(f_{y}\) is also continuous at \(\left ( x_{0},y_{0},y_{0}^{\prime }\right ) \) and also \(f_{y^{\prime }}\) is also continuous at \(\left ( x_{0},y_{0},y_{0}^{\prime }\right ) \) then there is unique solution on interval that contains \(x_{0}\).