On initial value problem in theory of the second order differential equations

We consider the properties of the second order nonlinear differential equations $b''=g(a,b,b')$ with the function $g(a,b,b'=c)$ satisfying the following nonlinear partial differential equation
\begin{gather*} g_{aacc}+2cg_{abcc}+2gg_{accc}+c^2g_{bbcc}+2cgg_{bccc}+g^2g_{cccc}+(g_a+cg_b)g_{ccc}- \\ 4g_{abc}-4cg_{bbc}-cg_{c}g_{bcc}-3gg_{bcc}-g_cg_{acc}+4g_cg_{bc}-3g_bg_{cc}+6g_{bb}=0. \end{gather*}
Any equation $b''=g(a,b,b')$ with this condition on the function $g(a,b,b')$ has the General Integral $F(a,b,x,y)=0$ shared with General Integral of the second order ODE's $y''=f(x,y,y'')$ with the condition $\frac{\partial^4f}{\partial y^{\prime4}}=0$ on the function $f(x,y,y')$ or $y''+a_1(x,y){y'}^3+3a_2(x,y){y''}^2+3a_3(x,y)y'+a_4(x,y)=0$ with some coefficients $a_i(x,y)$.