\hK(^SrSSKJr SSKJrJr SSKJr SSKJ r S Sjr Sr S S S .S jjr g)zUtility functions for classifying and solving ordinary and partial differential equations. Contains ======== _preprocess ode_order _desolve )Pow) Derivative AppliedUndef)Equality)WildNc[U[5(a'URR(a URnUR [ 5nU(dg[5R"UVs/sHoDR [5PM sn6n[U5S:wa[SU-5eUR5n[UR5nUcX4$UVs/sH^oBRS5(a5URU5(d[UR 5U-(dMMXDR#54PM` nnUR%U5nX4$s snfs snf)aPrepare expr for solving by making sure that differentiation is done so that only func remains in unevaluated derivatives and (if hint does not end with _Integral) that doit is applied to all other derivatives. If hint is None, do not do any differentiation. (Currently this may cause some simple differential equations to fail.) In case func is None, an attempt will be made to autodetect the function to be solved for. >>> from sympy.solvers.deutils import _preprocess >>> from sympy import Derivative, Function >>> from sympy.abc import x, y, z >>> f, g = map(Function, 'fg') If f(x)**p == 0 and p>0 then we can solve for f(x)=0 >>> _preprocess((f(x).diff(x)-4)**5, f(x)) (Derivative(f(x), x) - 4, f(x)) Apply doit to derivatives that contain more than the function of interest: >>> _preprocess(Derivative(f(x) + x, x)) (Derivative(f(x), x) + 1, f(x)) Do others if the differentiation variable(s) intersect with those of the function of interest or contain the function of interest: >>> _preprocess(Derivative(g(x), y, z), f(y)) (0, f(y)) >>> _preprocess(Derivative(f(y), z), f(y)) (0, f(y)) Do others if the hint does not end in '_Integral' (the default assumes that it does): >>> _preprocess(Derivative(g(x), y), f(x)) (Derivative(g(x), y), f(x)) >>> _preprocess(Derivative(f(x), y), f(x), hint='') (0, f(x)) Do not do any derivatives if hint is None: >>> eq = Derivative(f(x) + 1, x) + Derivative(f(x), y) >>> _preprocess(eq, f(x), hint=None) (Derivative(f(x) + 1, x) + Derivative(f(x), y), f(x)) If it's not clear what the function of interest is, it must be given: >>> eq = Derivative(f(x) + g(x), x) >>> _preprocess(eq, g(x)) (Derivative(f(x), x) + Derivative(g(x), x), g(x)) >>> try: _preprocess(eq) ... except ValueError: print("A ValueError was raised.") A ValueError was raised. z5The function cannot be automatically detected for %s. _Integral) isinstancerexp is_positivebaseatomsrsetunionrlen ValueErrorpopargsendswithhas variablesdoitsubs) exprfunchintderivsdfuncsfvarsrepseqs M/var/www/auris/envauris/lib/python3.13/site-packages/sympy/solvers/deutils.py _preprocessr%st$ HH ! !99D ZZ #F  VDVggl3VDE u:?13789 9yy{  NE |z#) 56a{1K1K EE$KK3q{{+e3 QM6D 5 4B 8OE 5s1EA E/Ec^[ST/S9nURU5(ag[U[5(awURST:Xa[ UR 5$URSRn[ UR 5nU(aU[U4SjU55- nU$UR(a[U4SjUR55$S$)a Returns the order of a given differential equation with respect to func. This function is implemented recursively. Examples ======== >>> from sympy import Function >>> from sympy.solvers.deutils import ode_order >>> from sympy.abc import x >>> f, g = map(Function, ['f', 'g']) >>> ode_order(f(x).diff(x, 2) + f(x).diff(x)**2 + ... f(x).diff(x), f(x)) 2 >>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), f(x)) 2 >>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), g(x)) 3 a)excluderc3<># UHn[UT5v M g7fN ode_order.0_rs r$ ode_order..s;d)At,,dc3<># UHn[UT5v M g7fr*r+r-s r$r0r1s9y!9Q%%yr2)rmatchr rrrrmax)rrr'rrvs ` r$r,r,^s. S4&!A zz!}}$ ## 99Q<4 t~~& &99Q<$$DT^^$Bc;d;;;I=AYYs9tyy99MAMT)prepc [U[5(aURUR- nU(dUc[ X5upSnUR SS5nUR S5nUR S5n UR SS5n UR S5n US :XaSS KJn Jn U nS nS nOUS :XaSSK J nJn UnSnSnUR SS5(a W"XSX8U XX%S9 nOUR SSUX&SSUS05nUS(dUW ;aUS:wa[SU-5eUUS;a#US:wa[W[U5-S-U-5eUSS:Xa#[[U5S-[U5-5e[WS-S-[U5-5eUS:Xa[XUUSUXZSUSUUSXXS9$US;a0n[U51S k- nUS!:XacUH;nUR!S"5(dMUR#UR%S"55 M= S#HnUU;dM UR#U5 M UH n[XUUXUSU USUUUS$9 nUUU'M" SUS%'UUS&'U$UW ;a[SU-5eUU;a[W[U5-S-U-5eUUS'UR'XS'.5 U$)(aThis is a helper function to dsolve and pdsolve in the ode and pde modules. If the hint provided to the function is "default", then a dict with the following keys are returned 'func' - It provides the function for which the differential equation has to be solved. This is useful when the expression has more than one function in it. 'default' - The default key as returned by classifier functions in ode and pde.py 'hint' - The hint given by the user for which the differential equation is to be solved. If the hint given by the user is 'default', then the value of 'hint' and 'default' is the same. 'order' - The order of the function as returned by ode_order 'match' - It returns the match as given by the classifier functions, for the default hint. If the hint provided to the function is not "default" and is not in ('all', 'all_Integral', 'best'), then a dict with the above mentioned keys is returned along with the keys which are returned when dict in classify_ode or classify_pde is set True If the hint given is in ('all', 'all_Integral', 'best'), then this function returns a nested dict, with the keys, being the set of classified hints returned by classifier functions, and the values being the dict of form as mentioned above. Key 'eq' is a common key to all the above mentioned hints which returns an expression if eq given by user is an Equality. See Also ======== classify_ode(ode.py) classify_pde(pde.py) NFtypexietax0rnode) classify_odeallhintszODE pde) classify_pderAzPDE pclassifyT)dicticsr;r<r>r=rr8rdefaultr4orderzHint not recognized: ordered_hintsz does not match hint z, is not a solvable differential equation in solvez: Cannot solve ) rHrsimplifyr8r=rFrJr4r;r<r>r:)all all_Integralbest>rJrIrKrOr )1st_homogeneous_coeff_best1st_power_series lie_group2nd_power_series_ordinary2nd_power_series_regular) rHrr=rMr8rFr>rJr4r:rNr#)rr#)r rlhsrhsr%getsympy.solvers.oder@rAsympy.solvers.pderDrstrNotImplementedError_desolverrremove removesuffixupdate)r#rrrHrMr8kwargsr:r;r<r=termsr@rA classifierstringdummyrDhintsretdictgethintsiksols r$r]r]s.R"h VVbff_ t|r( ::fd #D D B **U C D! B JJsOE u}<!  <!  zz*d##2$CC T.  6%t /#VG_67   x DI$54t;< < / /DI4EVc"g-0GG$NO O7^q BHH3t9TV V&ego8I&ICPRG&ST T yci0@8eGn!% "23uY Y 0 0u: EE > !::k**OOANN;$?@V=OOA&VA21UY%uW~U1XTXZCGAJ  X 04788 U #b'),CCdJKKf  LL$)* Lr7)Nr )NrINT) __doc__ sympy.corersympy.core.functionrrsympy.core.relationalrsympy.core.symbolrr%r,r]r7r$rrs4 8*"K\%NPKTKr7