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This ODE has the following adjoint
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This adjoint equation is in turn solvable by dsolve
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Now the solutions to the adjoint equation are integrating factors of the original LODE, so the two independent solutions implied in the general solution above
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are integrating factors of ode. These integrating factors could also be found using the intfactor directly
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Constructing solutions using integrating factors
How are these integrating factors transformed into a solution to the original problem? By using them to construct two first integrals; that is: two ODEs of lower order (in this case two first order ODEs). For that purpose it is provided the firint command which receives an exact ODE and returns a first integral. The idea is simple: an exact ODE is a total derivative - say dR/dx; firint returns the R + _C1:
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Eliminating y' from these two first integrals (and replacing _C1 by _C2 in one of them) leads to the solution f(x,y(x),_C1,_C2) = 0 to this ode. So this process could be run interactively, as shown, or in one step
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