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 \(x_{0}\).