The general first order linear ODE and its integrating factor

A related first integral can now be obtained either using the option , as in: firint(L_ODE, _mu = Mu), or multiplying directly L_ODE by Mu (by construction, Mu*L_ODE is exact)

The most general first order ODE reducible by an integrating factor depending only on  is a Linear ODE (see redode):

An exact nonlinear ODE which also has integrating factors of the form

Two integrating factors and the related first integrals

The answer to ode2 can be built from these two first integrals interactively, or by calling dsolve with extra arguments, indicating the use of the integrating factor scheme :

A third order ODE, an integrating factor for it, and the related first integral.

The firint command treates linear and non linear ODEs in equal footing, unless the optional method = formal, valid only for linear equations, is indicated. Consider

Calling firint without optional arguments will result in an error message telling ode4 is not exact. Passing method = formal, you instead get a complete set of independent first integrals obtained by first computing a basis of the solution space

A textbook example: the most general 2nd order non-homogeneous linear ODE having for solution basis  and , for the homogeneous part, and  as particular solution of the whole non-homogeneous equation, so the basis of solutions is

Verify that Basis is a complete basis for the solutions of ode5: from Basis construct the general solution then test it using odetest

A complete set of first integrals for ode5 computed from Basis

Verify this result