Solving Exact ODEs - Maple Help

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Solving Exact ODEs

Description

 • The general form of the exact ODE is given by:
 > exact_ode := D[2](C)(x,y(x))*diff(y(x),x)+D[1](C)(x,y(x))=0;
 ${\mathrm{exact_ode}}{:=}{{\mathrm{D}}}_{{2}}{}\left({C}\right){}\left({x}{,}{y}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{{\mathrm{D}}}_{{1}}{}\left({C}\right){}\left({x}{,}{y}{}\left({x}\right)\right){=}{0}$ (1)
 where C is an arbitrary function of its arguments. See Kamke's book, p. 28. This type of ODE can be solved in a general manner by dsolve, and the infinitesimals can also be determined by symgen.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor},\mathrm{symgen},\mathrm{symtest}\right)$
 $\left[{\mathrm{odeadvisor}}{,}{\mathrm{symgen}}{,}{\mathrm{symtest}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{exact_ode}\right)$
 $\left[{\mathrm{_exact}}{,}\left[{\mathrm{_1st_order}}{,}{\mathrm{_with_symmetry_\left[F\left(x\right),G\left(y\right)\right]}}\right]\right]$ (3)
 > $\mathrm{ans}:=\mathrm{dsolve}\left(\mathrm{exact_ode}\right)$
 ${\mathrm{ans}}{:=}{C}{}\left({x}{,}{y}{}\left({x}\right)\right){+}{\mathrm{_C1}}{=}{0}$ (4)

Implicit or explicit results can be tested using odetest

 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{exact_ode}\right)$
 ${0}$ (5)

A pair of infinitesimals for exact_ode are given by

 > $\mathrm{sym}:=\mathrm{symgen}\left(\mathrm{exact_ode}\right)$
 ${\mathrm{sym}}{:=}\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}\frac{{1}}{\frac{{\partial }}{{\partial }{y}}{}{C}{}\left({x}{,}{y}\right)}\right]$ (6)

Symmetries can be tested as well using symtest

 > $\mathrm{symtest}\left(\mathrm{sym},\mathrm{exact_ode}\right)$
 ${0}$ (7)
 See Also DEtools, odeadvisor, dsolve, and ?odeadvisor, where is one of: quadrature, linear, separable, Bernoulli, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, Chini, Riccati, Abel, Abel2A, Abel2C, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,types.