Use of group analysis in solving overdetermined systems of ordinary differential equations (Q1824742)

The authors consider an overdetermined system of two differential equations of orders m and n resp. (m\(\leq n)\) \(f(x,y,y',...,y^{(m)})=0\), \(g(x,y,y',...,y^{(n)})=0\). Each member of the set D of common solutions lies on the surface S defined by the intersection of the surfaces \(f(z_ 1,z_ 2,...,z_{m+2})=0\), \(g(z_ 1,z_ 2,...,z_{n+2})=0\) where \(z_ 1=x,z_ 2=y,z_ 3=y',...,z_{n+2}=y^{(n)}\). By the assumption that each of the equations is invariant under the same r-parameter \((\epsilon_ 1,...,\epsilon_ r)\) solvable Lie group of point transformations \(G^{(r)}\) \[ x^*=X(x,y;\epsilon_ 1,\epsilon_ 2,...,\epsilon_ r),\quad y^*=Y(x,y;\epsilon_ 1,\epsilon_ 2,...,\epsilon_ r) \] (1\(\leq r\leq m)\), there exist differential invariants \(u(x,y,y',...,y^{(r-1)})\), \(v(x,y,y',...,y^{(r)})\) such that the above equations reduce to the equivalent overdetermined system of equations \[ F(u,v,v',...,v^{(m-r)})=0,\quad G(u,v,v',...,v^{(n- r)})=0 \] for some functions F and G. By group analysis the authors establish that the surface S containing a set of common solutions D is a surface of dimensionality at most \(m+1-r\) in \((z_ 1,z_ 2,...,z_{n+2})\)-space. Supposing a curve \(v=\phi (u)\) solves the last system, then any solution of equation \(v(x,y,y',...,y^{(r)})=\phi (u(x,y,y',...,y^{(r-1)}))\) is a common solution of the initial system. Since u and v are invariants of \(G^{(r)}\) it follows that \(v=\phi (u)\) is invariant under the r-parameter solvable group \(G^{(r)}\). Hence \(v=\phi (u)\) can be reduced constructively to r quadratures. Thus they obtain explicitely a function \(\psi (x,y;c_ 1,c_ 2,...,c_ r)\) \((c_ 1,c_ 2,...,c_ r\) are constants) for which the equation \(\psi (x,y;c_ 1,c_ 2,...,c_ r)=0\) defines an implicit common solution of the initial system. Examples are given.