+ ir^RIHt^RIHt^RIHt^RIHtHtH t ^RI H t ^RI H t ^RIHtHt^RIHtHt^R IHt^R IHt^R IHt^R IHt^R IHt^RIHt^RI H!t!^RI"H#t#H$t$H%t%H&t&H't'H(t(H)t)H*t*H+t+H,t,H-t-H.t.H/t/H0t0H1t1H2t2]!R4t3]!R4t4Rt5Rt6Rt7Rt8Rt9Rt:Rt;RtRt?Rt@RtAR tBR!tCR"tDR#tER$tFR%tGR&tH]!R'4tIR(#)))randint)Function)Mul)IRationaloo)Eq)S)Dummysymbols)explog)tanh)sqrt)sin)Poly)ratsimp) checkodesol)slow)riccati_normalriccati_inverse_normalriccati_reduced match_riccatiinverse_transform_poly limit_at_infcheck_necessary_conds val_at_infconstruct_c_case_1construct_c_case_2construct_c_case_3construct_d_case_4construct_d_case_5construct_d_case_6rational_laurent_series solve_riccatifxcD\\V)V4\^V44#)rr)maxints&b/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/solvers/ode/tests/test_riccati.py rand_rationalr-s GVGV,ga.@ AAcv\\V^,4Uu.uFp\V4NK upV4#uupir))rranger-)r'degreer+_s&&& r, rand_polyr3!s. fQh@1v&@! DD@s6c\^V4p\^V4p\WV4p\WV4pV\^V48Xd\WV4pKWV, #r))rr3r)r'r1r+degnumdegdennumdens&&& r,rand_rational_functionr9%sT Q F Q F Av &C Av &C a 6* 9r.cTpVPV4p\V^^4p\V^^4pV^8Xd\V^^4pK\WEV,, Wc^,,, 4p\VP4WuV,,Wb^,,,4p\W#4p \ W4R8XgQhWWV3#)r*)Tr)diffr9rr r) ratfuncr'yfyypq1q2q0eqsols &&& r,find_riccati_oderE/sA B 1a (B 1a (B ' #Aq! , dR1W$ %B BGGIrrEzB1uH, -B R)C r 9 ,, , 2>r.cD\\^, , \^,^,^, \,\\^,)\^, , \\^,^, \^4^, ,,^, , ^^\,, , 3^\,^,^\,^,, ^^\,, \^,, ^\,R\,^\,^,,^\,^,, ^^\,, \^\^,RR7, , ^^\,, , 3R^\^,,^, , ^^\, ^\,^, , ^\, ^\,^, ^\^,,^, ,, ^\,^, \R^\, RR7^\,^, ^,, ^^\,^, , , ,\^^\, RR7, , 3\^\,^ , ^ \,^ ,, \^,^\,^ , , \^,)^\,^ , , ^\,^ , \^^ \,^ ,RR7, , ^\,^ , R\^,,^\,^ , ^,, ^\^,,^\,^ , , ,,^\^,,, , 3.pVFDwrr4V\V\W#48XgQhV\ V\W#4P 48XdKDQh R\,^, ^\^,,^\,,^, , R\, \)^, ^\,, ^\^,,^^\,, \)^, ^\^,,, ,,\)^, ^,, ^\)^, , ,R\,R^\,, \)^, ^\^,,, , ,\)^, , R\,^, \)^, ,^\,^\^,,^\,,^, ,, , ^\, ,3^^\^,,, R\, \)^, ^\,, ^\^,,^^\,, \)^, ^\^,,, ,,\)^, ^,, ^\)^, , ,R\,R^\,, \)^, ^\^,,, , ,\)^, , ^\, ,\^\)^, RR7^\^,,, , 3.pVFFwrr5pV\V\W#48XgQhV\ V\W#V4P 48XdKFQh R#) aN This function tests the transformation of the solution of a Riccati ODE to the solution of its corresponding normal Riccati ODE. Each test case 4 values - 1. w - The solution to be transformed 2. b1 - The coefficient of f(x) in the ODE. 3. b2 - The coefficient of f(x)**2 in the ODE. 4. y - The solution to the normal Riccati ODE. FevaluateN)r'r rrrcancel)testswb1b2r>bps r,test_riccati_transformationrT?sB 1q5 A1 Q Aq1u 1a46AaDF?+A--1Q37  1q1Q37 QqS1q5 ! 1acAg!a A!Gc!QUU.K#LLqRSTURUwV  AadFQJ Q1q Q!A#'AadFQJ'(AaC!Gc"a!ee6TVWXYVY W V71Q37 7,!!QUU;,= =  1rBqD1H 1acAg AqsQw1Q38c!RTAX&FGG1Q3QR7UWXY[\X\U\^_`a^a _ ^VAqD&!A#'"V#K$%&q!tVK- - ) E6 rN1a0000*1a8??AAAA AAadFQqSL1$% 1 a!A# !Q$1Q3A26AadF++,qb1fq[81qb1f:E 1b!A#h1"q&1QT6**+aR!V41qA267JAaCQRSTVWSWQWZ[\]Z]Q] R M 8 !   1QT6  1 a!A# !Q$1Q3A26AadF++,qb1fq[81qb1f:E 1b!A#h1"q&1QT6**+aR!V4qs:SQBFUZ=[]^_`bc_c]c=dd  E "rqN1a0000*1a<CCEEEE"r.c l\\4P\4\^,, \\\4,, \\\4^,,, \\4P\4\\4^,,\^,,\^,^, , ^^\^,,, , 3^\,^\,^ ,, \\4P\4,\^,\\4^,,\, , R\^,,^\, \)^, \^,, ,^,,^\)^, ^,,, \^\)^, RR7^\,^ ,, ,\\4^,,\\4P\4,R\^,\, ,\\)^, ,, , 3\\4^,\\4P\4,\^, \\4,\)\ ^4^, , , , ^\,^, ^,)^^\,^,^,,, ^\,^, ^\,^,^,, ,\\4^,,\\4P\4,^^\,^,, , 3\\4P\4\\4^,\, , \\4^,\\4P\4,^^\^,,, ,3R\^,)\, ^,,\^,^\,,^,, \\4P\4,\\4^,\, ,\\4^,\\4P\4,^\^,,\^,^\,,^,, ^\,\^,^\,,^,, ,^\^,^\,,^,, , \, ,^^\^,,, ,3^\,^\,^ ,, \\4P\4,\^,\\4,\, , R3\\4\\4P\4,^\, , \\4^, ,\\4^,\^,^, , ,R3.pVF!wrV\ V\\48XdK!Qh R#)z This function tests the transformation of a Riccati ODE to its normal Riccati ODE. Each test case 2 values - 1. eq - A Riccati ODE. 2. normal_eq - The normal Riccati ODE of eq. FrGNrJ)r&r'r;rr r)rOrC normal_eqs r,test_riccati_reducedrXsJ ! ! q!ta!f$q1qy0 ! ! qtQwA%1Q.AadF; !QqS1W ! ! $AqtQwq'88 1a41Q1}$q((!aR!VaK-83q Q< !"1q<* *,-aD!G 467diil C Q >ArAvJ ' ( !a!A$))A,!a%1rAaDF{!;; A#'A q!A#'A~&!A#'AaC!Ga<)?? !a A$))A, !"AaC!G - ! ! qtQwqy  !a!A$))A,AadF+ QTEAIMAqD1Q3JN+adiil:QqT1WQYF !a!A$))A,!AqD&!Q$1*q."9AaCA A#BB=#1qs Q'#()*"+ +-.!Q$Z 8 !QqS1W ! ! $Aqt|A~5  !QqTYYq\AaC!A$q&(1Q47AqD1H+== G' EP OB15555r.c \\4P\4R\^,,R\^,,, ^N\,, ^\,^\^,,R\^,,, R\^,,,^\,,^H, , , ^, \\4^,^\,^,, , \^4^, \, \\4,\^4^, ^\,^, , , , R^-\^,,^\^,,^i\^,,, ^^\^,,,^\,,^, , ^b\^,,^\^,,^i\^,,, ^^\^,,,^\,,^, , , ^\,^Q\^,,R\^,,, R\^,,,^<\,,^, , , ^,^\^\^,,R\^,,, R\^,,,^\,,^H, , ,\ R^^\,, RR 7^ \,^, , ^^\,^,, 3\\4P\4^\,^, ,\^, ^, \\4^,,, ^\,^, ^,\\4,^\,^,, , \^ 4^, , R \^,,R \,,^,R \^,,^\^,,,, , RR\,^, \^ 4^, ,^^\^,,^\^,,,, ,^/^$\^,,^\,,, ,R R \,^,, ,\ RR\,^, RR 7^ \,^,, \^, ^, 3\\4P\4R \^,,R\^,,, R\^,,,R\^,,, ^\,,^l, ^$\^,,^\^,,, R\^,,,R\^,,, R\^,,,^l\,, , , \^4^ , , \\^4^, , \\4,\^, \^4^, , , , \^, ^, \\4^,,^\,, , RR \^,,^$\^,,^\^,,, R\^,,,R\^,,, R\,,^l, , R\^,,^$\^,,^\^,,, R\^,,,R\^,,, R\,,^l, , , R\^,,^$\^,,^\^,,, R\^,,,R\^,,, R\,,^l, , ,R\,^ \^,,^H\^,,, ^\^,,,^\^,,, ^x\,,^$, , , \^4^ , ,^ ^\^,,^\^,,, ^5\^,,,^?\^,,, ^(\^,,,^ \,, , , ^^\^,,^\^,,, ^5\^,,,^?\^,,, ^(\,,^ , , ,\ R^^\,, RR 7\^, , \ R^ \, RR 7^ \,, 3\\4P\4\\^4^, ,\\^4^, ,^, , , \^,\\4,,\\\4^,,\\^4^, ,, ,R^^^3\\4P\4\ \^,4, \ \4\\4,,\\4\\4^,,,R^^^3\\4P\4\\\\4,4, \\4,\^,\\4^,,,R^^^3^\^,, \\4P\^4,^\,\\4P\4,, ^\\4,,R^^^3\\4P\4\^,, \^,\\4,,\^,\^,, \\4^,,,R^^^3\\4P\4\\4^,,\^,^, \^,^,, \\4,,^^\,^,, ,\\4^,,R^^^3. pVF;wrr4p\V\\4wrgWb8XgQhV'gK0W4V.V8XdK;Qh R#)z This function tests if an ODE is Riccati or not. Each test case has 5 values - 1. eq - The Riccati ODE. 2. match - Boolean indicating if eq is a Riccati ODE. 3. b0 - 4. b1 - Coefficient of f(x) in eq. 5. b2 - Coefficient of f(x)**2 in eq. iiriiNT;iFrGi iiDi0iiwilii7ihi$NrJrKrM) r&r'r;r rrr rrrr)rOrCresb0rQrRmatchfuncss r,test_match_riccatir_s  ! ! AqD3q!t8+bd2R7#ad( ad(;AX; #A;&(*;+, ,./ 023A$'1Q372C D Q46A:qt QqT!Vac!e^ , -  1a4AqD3q!t8#bAg-14q89Bq!tG AqD3q!t8 bAg %1 ,q 0=2 246qD"QT' AqD;q!t8; d;#%';(5) )+, -/13q!t8c!Q$h3F ad(4U44 0! ! BAaC%(!A#'2 1Q37 " ! ! qs2v1q!A$' 11QqSU 6 Q451q5 bE"H %(+AqD3q5(83(> ad(SAX (   1R!B%(Q!Q$1a4002r!Q$w Q$84 s1us{# $ B1q5)1Q373 !a  ! ! AqD3q!t8+c!Q$h6QTA a%1a4#ad(*SAX5AqD@ AqDq5 bE!G $'(1Q46z1Q4&71 !Q:' Q37AaD!G#QqS) *  AqD"QT'C1H$s1a4x/#ad(:SUB   1a4xAqD3q!t8+c!Q$h6QTA A 1a4xAqD3q!t8!3c!Q$h!> 1B " 1u"" "!eR1Wr!Q$w%6QT%A AqD&q5&& "#2q  ),.qAv1a4/? 1a40QT'0q!tG0$&(d0+,,  , /1!AqD&2ad72B QT'3q!tG3 d3#%'3(/)  ) BAaC%(!a%0 BA&!,%, ! ! q1Q46{A!QK!O44q!tAaDy@ !A$' 1qtAv;  q!Q ! ! s1a4y 3q6!A$;.Q!a? q!Q ! ! tAQK((1Q4/!Q$qtQw,> q!Q QT1Q499Q?"QqS11%551Q4? q!Q ! ! q!tad1Q4i'1a4Q<1q*@@ q!Q ! ! QqT1W1q1a4!84QqT99Aqs @ =  tQw  q!QWO E` %$RA. || 3B<5( (( %r.c  \^ \^,,^\^,,,^ \,, ^,\4\R\^ ,,\^ ,, ^\^,,,^\^,,,^ \^,,,^\^,,,^\^,,, ^ \^,,,^\^,,, ^\,,^ ,\4^3\^\4\R\^,,^\^,,,^\^,,,^\,, ^, \4^3\R\^,,^\^,,, ^\,,^, \4\R\^,,^ \^,,,^\,, ^ , \4^3\^ \^,,^ \^,,, ^ \^,,, ^\^,,,^\^,,,\^,, \^,,^ \,, \4\R\^,,\,\4R3\^\^,,^ \^,,,^ \^,,, ^ \^,,, \^,,^\,, ^,\4\^\^,,^\^,,,^\^,,, ^\,,^ ,\4R3.pVFwrp\W\4V8XdKQh R#)z This function tests the valuation of rational function at oo. Each test case has 3 values - 1. num - Numerator of rational function. 2. den - Denominator of rational function. 3. val_inf - Valuation of rational function at oo NirLrIirM)rr'r)rOr7r8vals r,test_val_at_infrcs R1WqAv 1 $q (!, SBYA !Q$ &1a4 /"QT' 9AadF BQq!tV KbQRTUQUg UXYZ[]^Z^X^ ^abcdad dgi iklm  Q  R1WqAv 1a4 '!A# - 2A6  R1WqAv ! #a '+ R1Wr!Q$w 1 $q (!,  R1Wr!Q$w AqD (1QT6 1AadF :QT AAqD H2a4 OQRS SAX\1  Qq!tVa1f_r!Q$w &1a4 /!Q$ 61 1/x in rational functions represented using Poly. Ni)r'as_numer_denomrrsubsrN)fnsr&er7r8s r,test_inverse_transform_polyrsUs 1WqAv!a"Q$(+AX1a4"QT'!Bq!tG+bd2R7#ad(SAX:MPRSTPT:TWY:YZAX1a4"QT'!Bq!tG+bd2R7"QT'Bq!tG:KbQRTUQUg:UXZ[\X\:\_a:ab1Wr!Q$wAqD 2ad7*R1W4s1uAqD H3q5 PSU UWXY SAXAqD 3q!t8 +c!Q$h 6ad BT!Q$Y NQUVWQW WZ] ]_`a !Q SAXAqD 3q!t8 +c!e 3b 8!< SAXAqD 419 ,s1u 4s :A> #s 5 E> #Ca(C///r.c \R\^,,^\^,,,^\,,^, \RR7\^\^,,^\^,,,^ \^,,,^ \^,,,^\^,,,^ \^,,,\RR7\^4\^4^, \^4\,^, ,.\^4^, \^4\,^, , ..3\R\^,,R\^,,,R\,,R,\RR7\^\^,,R\^,,, R\^,,,R \^,,, R \,,^1,\RR7\^4^, \^4^, \R 4R , ,.\^4^, \R 4R , , ..3\^\,^,\RR7\^\^,,^^\^4,, \^,,,^^ \^4,, \^,,,^^\^4,, \,,^\^4,,^,\R R7\^4\^4,^, \^4^, \\ ^^\^4,^,RR7^\^4,^,,, ^,4^, ,.\^4^, \\ ^^\^4,^,RR7^\^4,^,,, ^,4^, , ..3.pVFwrr4\ W\V4V8XdKQh R#)a> This function tests the Case 1 in the step to calculate coefficients of c-vectors. Each test case has 4 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. c - The c-vector for the pole. T extensioniii0i0iiieiijizz QQdomainFrGNrV)rr'r rrrr)rOr7r8polecs r,test_construct_c_case_1rs  R1WqAv ! #a 'd; Qq!tVbAg !Q$ &AqD 01QT6 9Bq!tG CQRVW ! A$q&4719Q;  !A$q&4719Q;"6!78  T!Q$Yad "SU *S 0!tD SAXAqD 419 ,tAqDy 83q5 @2 EqTXY !Q A$q&4>#% % &1a$x.2D)D(EF  QqS1Wa4( R1WBtAwJ1, ,RQZA/E EQtTUwYXYHY YQi  7 1Q A$q&4Aqay1}u=r$q'zBORSSTUVV V W qT!Vd3q!DG)a-%@"T!W*r/RUVVWXYY Y Z \  E*#$!#At4999#r.c \^\RR7\\^, ^,\^, ,\RR7^^\)R \, ,^, .\R \,,^, ..3\^\^,,^ \^,,, ^\^,,, ^,\RR7\^\,^, ^,\^,^,,\RR7\^4^, ^\^Y4)^b, .\^ 4)^b, ..3\\^,\^,, ^\,,\RR7\\^, ^,\^,^,,\RR7^^^\ ^4,\^<4R, ^\ ^4,^, , ,^ , ^\ ^4,^, .R \ ^4,\^<4R, ^\ ^4,^, ,,^ , R \ ^4,^, ..3\^\^,,\^,,^,\RR7\^\,^,^,\^,,\RR7\^4)^, ^^\ R4,\ R4)^, \^74R, , ,R, \ R4R, .R \ R4,\ R4^, \^74R, , ,R, \ R4)R, ..3\\^,^,\RR7\^\,^, ^,\^,^,,\RR7\^4^, ^^\ ^B4,\ ^B4)^6, \^4R, , ,^, R \ ^B4,R, \ ^B4^, .R\ ^B4,\ ^B4^6, \^4R, , ,^, ^\ ^B4,R, \ ^B4)^, ..3\\^,^ ,\RR7\\\ ^4, ^,\RR7\ ^4^\ ^4\^4^, ^\ ^4,, ,^, \ ^4^, \ ^4.\ ^4)\^4^, ^\ ^4,,,^, \ ^4)^, \ ^4)..3.pVFwrr4p\ W\W44V8XdKQh R #)ae This function tests the Case 2 in the step to calculate coefficients of c-vectors. Each test case has 5 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. mul - The multiplicity of the pole. 5. c - The c-vector for the pole. TriWii8i$iNrJrMii)rr'rr rr)rOr7r8rmulrs r,test_construct_c_case_2rs^" QT" a!eaZQ d3 1 "b1f+a-1b1f:a<.)  Qq!tVbAg !Q$ & *A> acAg\1q51* $a48 !Q R5&)!uRxj!  QTAqD[1Q3 T2 a!eaZQ "A6 1 DG)QrU3Y471, -b 0!DG)A+ > DGQrU3Y471, -b 0"T!W*Q,? A  Qq!tVad]Q T2 acAg\1q5 !15 1a d3i-$s)C!B%+5 6s :DIcM J d3ic32u4 5c 9DI:c>J L  QTAXqD) acAg\1a4!8 $a48 !Q T"X+Ry|afVm3 4R 7DHT9I4PR8TW< X T"XtBx{QsVF]2 3B 6$r( 4$r(SVW Y  QTBYT* a$q'kA qD1 Q r(AaDFQtBxZ' ( +T!WQYR A r(AaDFQtBxZ' ( +d1gXaZ$r(C E K+ EX#($Q!#At9Q>>>#(r.c,\4^..8XgQhR#)zT This function tests the Case 3 in the step to calculate coefficients of c-vectors. N)r rlr.r,test_construct_c_case_3rs  QC5 (( (r.c\\^,)^\^,,, ^\^,,,^\,,^,\RR7\^ \^,,^\^,,, ^ \,,^, \RR7^^ \,^, \^, R \,\^4^, \^, , ,^, .R \,^, \)^, ^\,\^4^, \^, ,,^, ..3\\^,)^ \^,,,^\^,,,^\^,,,^\^,,,^\,,^,\RR7\\^,^\^,,,^ \^,,,\, ^,\RR7^R \,\\)^\, ,^, .^\,\)\^\,,^, ..3\R \^,,\^,, \^,, ^\^,,, \^,, ^\,, ^, \RR7\^\^,,^ \,,^,\RR7^R\ ^4,\,R, R \ ^4,\,^$, \ ^4\,^, \ ^4)\,\R4R, ^\ ^4,\,^, , ,^, .R\ ^4,\,R, ^\ ^4,\,^$, \ ^4)\,^, \ ^4\,\R4R, ^\ ^4,\,^, ,,^, ..3\R \^,,^\^,,, ^\^,,, \^,, ^, \RR7\^ \,^, \RR7^^)\,^, ^\,^, \^, \)\^;4)R, \, ,.R\,^, R\,^, \)^, \\^;4)R, \,,..3\\^,)\^,, \^,, \^,, \, \RR7\\^,\RR7^R\,^, ^\,\)\\)R^\,, ,^, .^\,^, R \,\\)\R^\,,,^, ..3\\^,)\^,, ^\^,,, ^\^,,, \^,, \^,, ^\,,^, \RR7\^\,^, \RR7^^\ ^4,\,^, ^\ ^4,\,^, \ ^4\,^, \ ^4\,^, \ ^4)\,\^4)^, ^\ ^4,\,^, , ,^, .R \ ^4,\,^, R \ ^4,\,^, \ ^4)\,^, \ ^4)\,^, \ ^4\,\^4)^, ^\ ^4,\,^, ,,^, ..3.pVF6wrr4\ W\\ V^4p\WS^,4V8XdK6Qh R#)a This function tests the Case 4 in the step to calculate coefficients of the d-vector. Each test case has 4 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. mul - Multiplicity of oo as a pole. 4. d - The d-vector. Tri rziIBi iNrVrirLrMiiirrIra)rr'rr rr$rr!)rOr7r8rdsers r,test_construct_d_case_4rs@ adUQq!tV^a1f $qs *Q .TB Qq!tVa1f_r!t #a 'd; Q$r'1Q31afSj1Q3./1 2 QrA2a41afSj1Q3./12 4  adUQq!tV^a1f $qAv -!Q$ 61 !#Av.!333"r.c b\^\^,,\^,,\,^, \RR7\^ \^,,^\^,,,^\,,^, \RR7\^4^, \^4)^l, .\^4)^, \^4^l, ..3\^\^,,\^,,\^,, \^,,^\,, ^, \RR7\^ \^,,^\^,,,^\^,,,^\^,,,^\,,^,\RR7\^4^, R\^4,^, .\^4)^, ^\^4,^, ..3\\^,\, ^,\RR7\^\^,,^\,,^,\RR7\^4^, R\^4,^ , .\^4)^, ^\^4,^ , ..3.pVF/wrp\W\\^^4p\ V4V8XdK/Qh R#)z This function tests the Case 5 in the step to calculate coefficients of the d-vector. Each test case has 3 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. d - The d-vector. TrZZrNrMrI)rr'rr$rr")rOr7r8rrs r,test_construct_d_case_5rEs Qq!tVad]Q  "A6 Qq!tVa1f_qs "Q &T: q'!)d1gXc\ "d1gXaZa$=> Qq!tVad]QT !AqD (1Q3 . 2AdC Qq!tVa1f_qAv %!Q$ .1 4q 8!DI q'!)RQZ] #tAwhqj!DG)B,%?@ QTAX\1T* Qq!tVac\A q. q'!)RQZ\ "d1gXaZ471$=> E  !%c2q!<!#&!+++r.c \R\^,,^, \RR7\^\^,,^\^,,,^ \,,^,\RR7\^4^, \^, ,.\^4^, \^, , ..3\R\^,,^\^,,, ^\,, ^, \RR7\\^,\^,, ^\^,,,^\^,,, ^\^,,, ^\,, ^ ,\RR7^.^..3\R\^,,\^,,^ \,,^ ,\RR7\^\^,,^\^,,, ^\^,,, ^ \^,,, ^,\^,,, ^.\^,,, ^"\^,,, ^\,, ^*, \RR7^.^..3.pVFwrp\ W\4V8XdKQh R#)z This function tests the Case 6 in the step to calculate coefficients of the d-vector. Each test case has 3 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. d - The d-vector. rrNrMrI)rr'r rr#)rOr7r8rs r,test_construct_d_case_6rfs R1Wq[!D) Qq!tVa1f_r!t #a '48 A$q&1Q3,!A$q&1Q3,( R1WqAv ! #a '48 QTAqD[1QT6 !AadF *Qq!tV 3ac 9A =qN qc  R1Wq!t^bd "R '48 Qq!tVbAg 1a4 '"QT' 1Bq!tG ;bAg E1a4 O Q$t % qc  E" !!#A.!333r.c \\^,^\,, ^ ,\RR7\\^,\, \RR7\^4^^^^^RR^ RRR^ R R/3\^@\^,,R\,, R,\RR7\^@\^,,^P\^,,, R\^,,, R\,,R, \RR7\^ 4^, ^^^\R4R , ^\R 4R , R\R 4R , ^\R4R, /3\^\RR7\\^,R \^4,^, \^,,,^\^4,^ ,\^,,,R!^\^4,, \^,,,^^\^4,,\,,^, \RR7\^4^^^^\^4,^R^\^4,, ^\ RR^\^4,, RR7R\^4,, ^R^\^4,, R\^4,^,, ^\ RR^\^4,, RR7R\^4,^,, RR^\^4,, R\^4,^,, /3\\^,^\^,,, ^\^,,,^ \,,^ , \RR7\\^,^, \RR7\ ^^^^^^^^^^R^R^/3\^\^,,^\^,,,^\,, ^,\RR7\^ \^,,\^,, ^\^,,, ^\,,^,\RR7\^4^, ^^^\R4R, R\R4R, R\R4R, R\R"4R, R \R#4R, /3\R$\^,,^\,,^, \RR7\^\^,,^ \^,,,^\^,,,^\^,,,^\,,^ ,\RR7\ ^^^^R^R%\^G4)^1, R^RRR \^ 4^, /3.pVF wrr4rVV\ W\W4V48XdK Qh R#)&a This function tests the computation of coefficients of Laurent series of a rational function. Each test case has 5 values - 1. num - Numerator of the rational function. 2. den - Denominator of the rational function. 3. x0 - Point about which Laurent series is to be calculated. 4. mul - Multiplicity of x0 if x0 is a pole of the rational function (0 otherwise). 5. n - Number of terms upto which the series is to be calculated. Triii?iiiiKiil`B l-Ri0i}'FrGi i ii)l?PTl@qsl@4lZdj mNirJrMrarVrKr{lF41lJ!tzrrI)rr'r rrrr$)rOr7r8x0rnrs r,test_rational_laurent_seriesrs& QTAaCZ!^Q$/ QTAXqD) !a Aq"b!RRB3  R1WtAv  $a48 R1Wr!Q$w QT )DF 2S 8!tL !Q1 AhK !QtWS["a n[6P 1V9U?   QT" QTRQZ!^QT) )QtAwY^QT,A AS1TRSW9_VWYZVZDZ Z qay=!  !"d 4 QA AQKB47NAs2rAd1gI~PU/VXZ q'Y0QtAwYd1g(991c"b1TRSW9n_d? Q"check_dummy_sol..s32tt2sFTc3L<"TFwrVPVS4xK R#5i)N)dummy_eq)rs1s2 dummy_syms& r,rrs#I8Hfbr{{2y))8Hs!$N) isinstancer lhsrhsrr&r'r%r rpallrzip)rCsolserr2r^solsrrDs&&f r,check_dummy_solrs "b VVbff_RA&HA 1q )5 )D tB/3 4tHHR #tD 4 33[233333[23 3 33 3 3ID8HI333ID8HI I II I 5s:D:cD/\R4p\\\4P \4\\4^,,^, ^4\\\4\ ^44\\\4\ ^4)4.3\\4^,\\4P \4,^\\4,\, ,^\^,, ,\\\4R V,\, V\,\^,,, 4.3^\^,,\\4P \4,\^\\4,\\4P \4,^, ,, \\4^, \\4,,\\\4V^\^,,,V\,, 4.3\\\4P \4\\4^,)^\^,\^,, , , 4\\\4^\^,\, , 4.3\^,^\,^\, ,\\4,, \\4^,,\\4P \4,\\\4V\,\^,,^\,,V\^,,, 4\\\4\4.3\^,\\4P \4,\^,,\^\\4^,,\\4P \4,,, \\4,\\\4V\^,,\,V\^,,, 4\\\4\^,4.3\\4^,)\\4P \4,^\^,,^\,, ^,\^, ^,^\,^, ^,,, ,\\\4^ V,\,^V,, ^\^,,, ^<\^,,,^^\^,,, ^H\^,,,^\,, ^,^V,\^,,^ V,\,, ^V,,^\^,,,^\^,,, ^9\^,,,^:\^,,, ^\^,,,^\,, , 4\\\4^\,^, ^\^,,^\,, ^,, 4.3\\4^,\\4P \4,^\^,,^\^,,, ^ \^,,,^\^,,,^\^,,,^\,, ^,^\^,,, , \\\4^\^,,^\^,,, \^,, ^\^,,,^\,,^, ^\^,,^\^,,, , 4.3\\\4P \4\^,)^\^,,,^(\^,,, ^-\^,,,^\,, ^,\^ ,^ \^ ,,, ^B\^ ,,,^\^ ,,, R\^,,,R\^,,, R\^,,,R\^,,, R\^,,,^\^,,, ^B\^,,,^ \,, ^,, \\4^,,\\4,4\\\4^\^,^\^,,, ^\^,,,^\^,,, ^\^,,,^\,, ^,, 4.3\\\4P \4\\\4,^\,,^\,^, \\4^,,^\,^,, ,^\^,,^\,, ^,^\^,,^\^,,, ^,, ,\ ^4^, , 4\\\4^^\,, ^\,^, , 4.3\\\4P \4R!\^,,^0\,, ^, ^\^,,^(\^,,, ^\,,^,, ^\\4^,,^\,^,, ,4\\\4^\,^,^\,^, , 4.3\\4P \4^\^,,^,\\4^,,\, ,^\^,,\, ^,\\4,\\^, ,, ,^\^,,^\,, ^,\\^, ^,,, ,\\\4V)\^,, \^,,^\,, V\,V, \^,,\^,, \^,,\, , 4\\\4R"\^, , 4.3\\4P \4^\,\\4^,^,,\, , \\\4\)V,\\^,,,V\^,,, 4\\\4\)4.3\\\4P \4\\\4,\ ^4^, ^\,, , \^, \ ^4^, , \\4^,,^\,^, \ ^4^, , , ,\ ^4^, , R\^,,R\,, R,^\^,,^ \^,,, , ,4\\\4^ \, \, 4\\\4^(\^,,^\^ ,,,R\^ ,,,R \^ ,,,R \^ ,,,R \^ ,,,R \^,,,R \^,,,R\^,,,R\^,,,R\^,,,R\^,,,R\^,,,R\,,R,^\^,,^\^ ,,,R\^ ,,,R\^ ,,,R\^ ,,,R \^ ,,,R\^,,,R\^,,,R\^,,,R\^,,,R\^,,,R\^,,,R\^,,,R\,,, 4.3\\\4P \4^\^,,^\^,,,\)^, \ ^4^, , \\4^,,,^,4\\\4^\,4.3\\\4P \4R#\^,,^, ^\,^, ,\^, \ ^4^, , \\4^,,,^ ,^\, \\4,\, ,^\, ,4\\\4R#\,^, ^, 4.3.pVFwr#\W#V4K R#)$a This function tests the computation of rational particular solutions for a Riccati ODE. Each test case has 2 values - 1. eq - Riccati ODE to be solved. 2. sol - Expected solution to the equation. Some examples have been taken from the paper - "Statistical Investigation of First-Order Algebraic ODEs and their Rational General Solutions" by Georg Grasegger, N. Thieu Vo, Franz Winkler https://www3.risc.jku.at/publications/download/risc_5197/RISCReport15-19.pdf C0iiirurvryrxi0HiiFi4"i<ii2ҍi"0 iimi^rwi i#??!A$F AaD2ad7Q;ad+ ,b1q!tn= 1q1Q499Q<2ad7RT>A#5Q AaCED9#  AaD1R46AbD=2ad7*R1W4r!Q$w>AqDH Q$rT!Q$Y2a'!B$.1a47"QT'A 1a4QT'q!tG$&'c*+ ,-/!qsQw1a4 A#BB7.  !a!A$))A,!AqD&1QT6/Bq!tG";a1f"D adF#T## !!Q$"( ( AaD1QT6AadF?QT)AadF2QqS81QT ICPQSTPTH T 1H1a4x "%ad(+-/1W579!t<>?@ACDQ47KMNqTR S AaD!QTAadF]R1W,r!Q$w6AqD@1Q3FJK LM  1Q499Q<1Q4!A#1q!A$'(91Q37(CC q!tVac\B AqD2ad7!2Q!6 78:;A$q&A B AaD1qs7QqS1W% &' 1Q499Q<#ad(RT/B.AqD2ad71BQqS1H11LM!ai1q!" # AaD1Q37QqS1W% &' ! ! !Q$ AaD!G+A--1a4!a10Eq!KH1 q!tVac\A%1q51* 5 6 AaDB3A:1$qs*RTBYA-=1-Dq!t-K. adBAJ' ) ! ! qsAaD!GaK(** AaDA2b51QT6>BAI. /AaD1"> 1Q499Q<1Q41a!A#.!A#!Q,!a1G qSUQqT!V^2 tAv&),QTDF):S)@2ad7RPQSTPTWCT(UV W AaD1q5!) b11b52ae8(;c!R%i(G$qRTu*(T !R%K) A+)&(.q!t )46=adl)CELQPQT\)RT\ qDU)AqD=)!#,QT>)24=adN)CENq[)QS\)]BhQU"SBY.ae;eArEkIFSTVWSWKW 1a4K!!Q$,')0A68@A FHPQRTUQU VX` TYQTM"$,QJ/(12 3  1Q499Q<AqD2ad7*qbdQqT!VmQqT1W-DDqHI AaD!A# 1Q499Q<AqDRT!V+qsQqT!V|QqT1W.DDq5!A$,q.!#$Q3' ( AaD"Q$q&1* Ad EJ$r.c \R4p\\\4P \4^\, \\4,\^, , ^^ \,, \\4^,,^\,^ , , ,R\^,,R\^,,, R\,,R, ^\^,,^\^,,, R\^,,,R\,, R,, ,R,4\\\4^\,^,^\,^ , , 4.3.pVFwr#\ W#V4K R #) z This function tests the computation of rational particular solutions for a Riccati ODE. Each test case has 2 values - 1. eq - Riccati ODE to be solved. 2. sol - Expected solution to the equation. rii/ii-i*irN)r r r&r'r;rrs r,test_solve_riccati_slowrXs$ tB 1Q499QORUVWYZVZRZ>Z F??  ! " AaD2a4!8acAg& '(  E$r.N)Jsympy.core.randomrsympy.core.functionrsympy.core.mulrsympy.core.numbersrrrsympy.core.relationalr sympy.core.singletonr sympy.core.symbolr r &sympy.functions.elementary.exponentialr r%sympy.functions.elementary.hyperbolicr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrsympy.polys.polytoolsrsympy.simplify.ratsimprsympy.solvers.ode.subscheckrsympy.testing.pytestrsympy.solvers.ode.riccatirrrrrrrrrrr r!r"r#r$r%r&r'r-r3r9rErTrXr_rcrmrsr}rrrrrrrrrrrlr.r,rs%(00$".=698&*3% SM CL BE  ?FD36l`)F).X =2.+0\#:L;?|)<4~,B4@@GFJ$z%z%%r.