Geodesic
On initial value problem in theory of the second order differential equations
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2003), pp. 51-58.

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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 ∂4f∂y′4=0 on the function f(x,y,y′) or y″+a1(x,y)y′3+3a2(x,y)y″2+3a3(x,y)y′+a4(x,y)=0 with some coefficients ai(x,y). 
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Valerii Driuma; Maxim Pavlov. On initial value problem in theory of the second order differential equations. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2003), pp. 51-58. https://geodesic-test.mathdoc.fr/item/BASM_2003_2_a4/

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