On \(L^2\) solutions of third order nonlinear differential equations (Q2761976)

The authors investigate the third-order nonlinear differential equation NEWLINE\[NEWLINEy'''+ p(t) y''+ q(t) y'+ r(t) f(y, y',y'')= 0,NEWLINE\]NEWLINE where \(p\), \(q\), \(r\) are continuous functions on \([0,+\infty)\), \(f\) is continuous on \(\mathbb{R}^3\) and \(p\geq 0\), \(q\geq 0\), \(r> 0\), \(x_1f(x_1,x_2,x_3)> 0\) for \(x_1\neq 0\). They give conditions for certain classes of solutions to belong to \(L^2\). Conditions which ensure that certain types of solutions do not exist are given.NEWLINENEWLINENEWLINEThe main results are applied to the case where the coefficients are powers of \(t\).