Solving Linear Second Order ODEs for which a Symmetry of the Form [xi=0, eta=F(x)] Can Be FoundDescriptionExamples
<Text-field bookmark="info" style="Heading 2" layout="Heading 2">Description</Text-field>All second order linear ODEs have symmetries of the form [xi=0, eta=F(x)]. Actually, F(x) is always a solution of the related homogeneous ODE. There is no general scheme for determining F(x); see dsolve,linear).When a symmetry of the form [xi=0, eta=F(x)] is found, this information is enough to integrate the homogeneous ODE (see Murphy's book, p. 88).In the case of nonhomogeneous ODEs, you can do the following:1) look for F(x) as a symmetry of the homogeneous ODE;2) solve the homogeneous ODE using this information;3) set each of _C1 and _C2 equal to 0 and 1 in the answer of the previous step, in order to obtain the two linearly independent solutions of the homogeneous ODE;4) use these two independent solutions of the homogeneous ODE to build the general solution to the nonhomogeneous ODE (see Bluman and Kumei, Symmetries and Differential Equations, p. 132 and ?dsolve,references).