Solving Linear Second Order ODEs for which a Symmetry of the Form [xi=0, eta=F(x)] Can Be Found
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).
A nonhomogeneous ODE example
A nonhomogeneous example step by step
Steps 1) and 2) mentioned above
Step 3): two independent solutions for the homogeneous_ode
Step 4): a procedure for the general solution to the original nonhomogeneous ODE (ode) is given by
where s1 and s2 are the linearly independent solutions of the homogeneous ode (sol_1 and sol_2 above), F is the nonhomogeneous term (here represented by F(x)), and W is the Wronskian, in turn given by
from which the answer to the nonhomogeneous ODE follows
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