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Theory and Modern Applications

Table 1 Two groups of nonlinear ordinary differential equations

From: Conditional linearization of the quintic nonlinear beam equation

Group

Generators

Ordinary differential equations

1

\(X_{1}+X_{2}\), \(X_{1}+X_{2}+X_{3}\)

\(\begin{array}[t]{l} c\phi'+(m+P_{0})\phi''+EI\phi^{(4)}-3EI\phi ^{\prime\prime 3}+\frac{3}{2}P_{0}\phi^{\prime 2}\phi''\\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi^{\prime\prime 3}+\frac {9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \)

\(X_{1}+X_{2}+X_{4}\)

\(\begin{array}[t]{l} c+c\phi'+(m+P_{0})\phi''+EI\phi^{(4)}-3EI\phi ^{\prime\prime 3}+\frac{3}{2}P_{0}\phi^{\prime 2}\phi'' \\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi^{\prime\prime 3}+\frac {9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \)

2

\(X_{2}+X_{4}\), \(X_{2}+X_{3}+X_{4}\)

\(\begin{array}[t]{l} c+P_{0}\phi''+EI\phi^{(4)}-3EI\phi^{\prime\prime 3}+\frac {3}{2}P_{0}\phi^{\prime 2}\phi''\\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi^{\prime\prime 3}+\frac {9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \)

\(X_{2}+X_{3}\)

\(\begin{array}[t]{l}P_{0}\phi''+EI\phi^{(4)}-3EI\phi^{\prime\prime 3}+\frac {3}{2}P_{0}\phi^{\prime 2}\phi''\\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi ^{\prime\prime 3}+\frac{9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \)