First-order ODEs with curvature K(x,u)=κ(x) are integrable by quadratures exactly when the associated linear operator L=d²/dx²+κ(x) has a non-zero Liouvillian solution, with Kovacic's algorithm deciding the rational-κ case.
Olver.Equivalence, Invariants and Symmetry
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Noether symmetries of time-dependent damped nonlinear multidimensional wave equations produce conservation of linear and angular momentum, with the algebra enlarging to a conformal subalgebra for particular damping and nonlinearity forms.
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From curvature to Kovacic: a geometric approach to integrability of scalar ODEs
First-order ODEs with curvature K(x,u)=κ(x) are integrable by quadratures exactly when the associated linear operator L=d²/dx²+κ(x) has a non-zero Liouvillian solution, with Kovacic's algorithm deciding the rational-κ case.
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Noether symmetries and conservation laws of a class of time-dependent multidimensional nonlinear wave equations
Noether symmetries of time-dependent damped nonlinear multidimensional wave equations produce conservation of linear and angular momentum, with the algebra enlarging to a conformal subalgebra for particular damping and nonlinearity forms.