second order ordinary differential equation

A second order ordinary differential equation  $F(x,y,\frac{dy}{dx},\frac{d^{2}y}{d{x}^2})=0$  can often be written in the form

${\displaystyle \frac{d^{2}y}{d{x}^2}}=f(x,y,{\displaystyle \frac{dy}{dx}}).$(1)

By its general solution one means a function   $x\mapsto y=y(x)$  which is at least on an interval twice differentiable and satisfies

$$y^{\prime \prime }(x)\equiv f(x,y(x),y^{\prime }(x)).$$

By setting  $\frac{dy}{dx}:=z$,  one has  $\frac{d^{2}y}{d{x}^2}=\frac{dz}{dx}$,  and the equation (1) reads $\frac{dz}{dx}=f(x,y,z)$.  It is easy to see that solving (1) is equivalent (http://planetmath.org/Equivalent3) with solving the system of simultaneous first order (http://planetmath.org/ODE) differential equations

$\{\begin{array}{cc}\frac{dy}{dx}=z,\hfill & \\ \frac{dz}{dx}=f(x,y,z),\hfill & \end{array}$(2)

the so-called normal system of (1).

The system (2) is a special case of the general normal system of second order, which has the form

$\{\begin{array}{cc}\frac{dy}{dx}=\phi (x,y,z),\hfill & \\ \frac{dz}{dx}=\psi (x,y,z),\hfill & \end{array}$(3)

where $y$ and $z$ are unknown functions of the variable $x$.  The existence theorem of the solution

$\{\begin{array}{cc}y=y(x),\hfill & \\ z=z(x)\hfill & \end{array}$(4)

is as follows; cf. the Picard–Lindelöf theorem (http://planetmath.org/PicardsTheorem2).

Theorem.  If the functions $\phi$ and $\psi$ are continuous and have continuous partial derivatives with respect to $y$ and $z$ in a neighbourhood of a point  $(x_0,y_0,z_0)$,  then the normal system (3) has one and (as long as $|x-x_0|$ does not exceed a certain ) only one solution (4)which satisfies the initial conditions   $y(x_0)=y_0,z(x_0)=z_0$.  The functions (4) are continuously differentiable in a neighbourhood of $x_0$.