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

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

2002-07-19

arxiv.org/abs/nlin/0207035v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

8 pages, Latex

Scientific paper

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 $$ \frac{d^2 g_{cc}}{da^2}-g_{c}\frac{dg_{cc}}{da}-4\frac{dg_{bc}}{da}+ $$ $$ +4g_{c}g_{bc}-3g_{b}g_{cc}+6g_{bb}=0, $$ 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 $\frac{\partial^4 f}{\partial y'^4}=0$ on 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).

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