On Equation for Initial Values in Theory of the Second Order Ordinary Differential Equations

Abstract

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 $$\frac{d^2{g}_{cc}}{d{a}^2}-g_c\frac{d{g}_{cc}}{da}-4\frac{d{g}_{bc}}{da}+$$ $$+4g_c{g}_{bc}-3g_b{g}_{cc}+6g_{bb}=0,$$ where: $$\frac{d}{da}=\frac{\partial }{\partial a}+c\frac{\partial }{\partial b}+g\frac{\partial }{\partial c}.$$ Any equation b''=g(a,b,b') with this condition on 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 condition $\frac{{\partial }^{4}f}{\partial y^{\prime 4}}=0$ on function f(x,y,y') or $$y^{\prime \prime }+a_1(x,y)y^{\prime 3}+3a_2(x,y)y^{\prime 2}+3a_3(x,y)y^{\prime }+a_4(x,y)=0$$ with some coefficients a_{i}(x,y).

8 pages, Latex