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## Main Question or Discussion Point

A second order ode [itex]Py'' + Qy' + Ry = 0[/itex] is exact if there exists a first order ode [itex] Ay' + By [/itex] such that

[tex] (Ay' + By)' = Ay'' + (A' + B)y' + B'y = Py'' + Qy' + Ry = 0[/tex]

How can one cast the analysis of this question in terms of exact differential equations?

In other words, could somebody explain this interesting quote:

[tex] (Ay' + By)' = Ay'' + (A' + B)y' + B'y = Py'' + Qy' + Ry = 0[/tex]

How can one cast the analysis of this question in terms of exact differential equations?

In other words, could somebody explain this interesting quote:

Thanks!The derivation of the conditions of exact integrability of an ordinary differential equation of the nth. order (or of a differential expression involving derivatives of a single dependent variable with regard to a single independent variable) is sometimes made to depend upon the theory of integration of an expression, exact in the sense of the foregoing chapter. As however the connection is not immediate and this method is not the principal method, it will be sufficient here to give the following references to some of the writers on the subject, in whose memoirs references to Euler, Lagrange, Lexell, and Condorcet, will be found in ...

Forsyth - Page 33