
    g0                     j   d dl mZmZmZ d dlmZmZmZ d dlm	Z	 d dl
mZmZ d dlmZmZ d dlmZ d dlmZmZ d dlmZmZmZ d d	lmZ d d
lmZmZ d dlmZm Z  d dl!m"Z"m#Z#  ed          \  Z$Z%Z&Z'Z( edd          \  Z)Z*Z+Z, ed          Z- ed          Z. ed          Z/e#d             Z0d Z1dS )    )
DerivativeFunctiondiff)IRationalpi)Eq)Symbolsymbols)explog)sqrt)cossin)Eierferfi)Integral)checkodesolchecksysodesol)besseljbessely)raisesslowzC0:5zu,x:zT)realfghc                     t          t          d            t          t          d            t          t          t                                        t                    t          t          t          t                    t                              dt          t                                        t                     t          t          t                                        t                    z   dz
  fk    sJ t          t          t                                        t                    t          t          t                    t                              dusJ t          t          t                                        t                    t          t          t                    t                              dk    sJ t          t          t                    dz  dt          t                    z  z   d	t          t                    z  z
  t          z   d
          } t          t          | j        t                    |           dk    sJ t          t          | j        t                    t          t          t                              z  |           dk    sJ t          t          | j        t          d	          |           dk    sJ t          t          | j        t          d	          t          t          t                              z  |           dk    sJ t          t          | j        t          d          |           dk    sJ t          t          | j        t          d          t          t          t                              z  |           dk    sJ t          t          | j        t          d          t          t          t                    t          t          t                    z                      ddt          dz  z  t          t                    dz   d	z  t          t                    z   z  t          t                    dz   z  t          t                    d	z  z  dt          dz  z  t          t                    dz  z  z
  dz
  fk    sJ t          t          t          t          t                              t          z   t                    t          z  t          t          t          t                              t          z   d
                    dk    sJ t          t          t          t          t                              t          z   t                    t          z  t          t          t          t                              t          z   d
          d          dk    sJ t          t          t                                        t          d	          t          t          t                    t          t          t          z  z             t          t          t                    t          t          t          z  z             t          t          t                    t          t          z  t          t          d	z  z  z             g          dddt          fgk    sJ t          t          t                                        t          d	          t          t          t                    t          t          t          z  z             t          t          t                    t          t          t          z  z             t          t          t                    t          t          z  t          t          d	z  z  z             h          dddt          fhk    sJ t          t          t                                        t                    dt          t                    z  d	z  z
  t          t          t                    d	z  t                              ddgk    sJ t          t          t                                        t                    t          t                    z
  t          t          t          t                    z  t          t                                        dk    sJ t          t          t          t                    t          z            z  t          t                    z   t          t          t                                        t                    z  z
  }t          t          t                    t          t          t          t          z            t           z                      }t          ||          d                             t          t                              rJ t          t          t          t          t          t                    t                    z  t                    t          z  t          t                              }t          t          t                    t          t          t          t                    z  z   t          t                    z   t          t                    z
            }t          ||d	d          d
         sJ t          d	z  t          t                                        t          d	          z  t          t          t                                        t                    z  z   d	t          d	z  z  dz   t          t                    z  z   }t          t          t                    t          t!          dt"          z  t%          d	          t          z            z  t          t'          dt"          z  t%          d	          t          z            z  z             }t          ||          dk    sJ t          t          t                                        t                    t          t                    t)          t                    z             t          t)          t                                        t                    t          t                    t)          t                    z             g}t          t          t                    t           t          t          d	t          z            z  z             t          t)          t                    t          t          t          d	t          z            z  z             g}t          ||          dd
d
gfk    sJ d S )Nc            	          t          t          t          t                                        t                    t          t          t          t                    t                              S N)r   r   xyr   r	        b/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/solvers/ode/tests/test_subscheck.py<lambda>z"test_checkodesol.<locals>.<lambda>   s2    {1Q77<<??BqAwwNNKK r%   c            	          t          t          t                                        t                    t	          t          t          t
                    t                    t          t          t
                              S r!   )r   r   r"   r   r	   r#   r$   r%   r&   r'   z"test_checkodesol.<locals>.<lambda>   sF    {1Q4499Q<<AaGG: :Aww    r%   F   T)Fr)            r   )Tr      <      	   )solve_for_func)orderr1      )r   
ValueErrorr   r   r"   r   r	   r#   lhsr   r   C1C2hasr   r   r   r   r   r   r   )sol1eq3sol3eqnsoleqeqss          r&   test_checkodesolr@      s    :KKLLL
:     qttyy||R!Q^^44	11!QQ/!345 5 5 5qttyy||R!a[[11====qttyy||R!a[[11Z????addAg1Q44!AaDD&(1,a00DtDHa(($//9<<<<tDHa((QqTT2D99YFFFFtDHa++T22i????tDHa++C!II5t<<	IIIItDHa++T22i????tDHa++C!II5t<<	IIIItDHa++R!aAh-?-?@@	1a4#a&&1*q3q6612A
FFAI !!Q$s1vvqy 01345 	66 6 6 6 tC!IIM1--a/C!IIM11E1EFF   tC!IIM1--a/C!IIM11E1E  !*+ + + +qttyyAAaDD"r!t))<)<
1Q44bdR!bdR1Wn55)7 8 8	Ir{+, , , , qttyyAAaDD"r!t))<)<
1Q44bdR!bdR1Wn55)7 8 8	Ir{+, , , , qttyy||a!fQh.1Q447A??	I   qttyy||add*Br#a&&y!A$$,?,?@@IMMMM C!QKK-!A$$
1Q4499Q<<
/CaddCBqD		QB(())D3%%a(,,QqTT22222
Z*QqTT1---q11!3SVV
<
<C
QqTT23q66	>CFF*RUU2
3
3CsCq???BBBB	
AqttyyA	!QqTTYYq\\"2	2a1fbj!A$$5F	FB
QqTT2gac47719---71Q3Q	3J3J0JJ
K
KCr39,,,,addiillAaDD1Q44K(("QqTTYYq\\1Q44!A$$;*G*G
HCaddRC"S1XX+%&&1Q44bQqSk1A(B(B
CCsC  TAq6N222222r%   c                  ]   t          dt                    \  } }}t          d          }t          t	           | |          |          d ||          z            t          t	           ||          |          d | |          z            f}t           | |          dt
          z  t          dt          d          z  |z            z  dt          z  t          dt          d          z  |z            z  z             t           ||          dt          d          z  t
          z  t          dt          d          z  |z            z  dt          d          z  t          z  t          dt          d          z  |z            z  z             g}t          ||          d	d
d
gfk    sJ t          t	           | |          |          d | |          z  d ||          z  z             t          t	           ||          |          d | |          z  d ||          z  z             f}t           | |          dt
          z  t          |t          d           dz  t          dd          z   z            z  dt          z  t          |t          d          dz  t          dd          z   z            z  z             t           ||          t
          t          d           dz  t          dd          z   z  t          |t          d           dz  t          dd          z   z            z  t          t          dd          t          d          dz  z   z  t          |t          d          dz  t          dd          z   z            z  z             g}t          ||          d	d
d
gfk    sJ t          t	           | |          |           | |           ||          z             t          t	           ||          |          d | |          z  d ||          z  z             f}t           | |          t
          t          t          d          |z  dz            z  t          t          t          d          |z  dz            z  z   t          |t          dd          z            z            t           ||          t
          dz  t          d          t          z  dz  z
  t          t          d          |z  dz            z  t          d          t
          z  dz  t          dz  z   t          t          d          |z  dz            z  z   t          |t          dd          z            z            g}t          ||          d	d
d
gfk    sJ t          t	           | |          |           | |           ||          z   dz             t          t	           ||          |          d | |          z  d ||          z  z   dz             f}t           | |          t
          t          |t          d           dz   z            z  t          t          |t          d          dz   z            z  z   t          dd          z
            t           ||          t
          t          d           dz   z  t          |t          d           dz   z            z  t          dt          d          z   z  t          |t          d          dz   z            z  z   t          dd          z
            g}t          ||          d	d
d
gfk    sJ t          t	           | |          |           | |           ||          z   dz             t          t	           ||          |          d | |          z   ||          z   dz             f}t           | |          t
          t          t          d          |z            z  t          t          t          d          |z            z  z   t          |          z  t          dd          z
            t           ||          t          d          t
          z  t          t          d          |z            z  t          d          t          z  t          t          d          |z            z  z
  t          |          z  t          dd          z
            g}t          ||          d	d
d
gfk    sJ t          t	           | |          |          d|z   | |          z  d ||          z  z             t          t	           ||          |          d | |          z  d|z   ||          z  z             f}t           | |          t
          t          t          d|                                                    z  t          t          t          d|                                                     z  z   t          t          d|z  |                                                    z            t           ||          t
          t          t          d|                                                    z  t          t          t          d|                                                     z  z
  t          t          d|z  |                                                    z            g}t          ||          d	d
d
gfk    sJ t          t	           | |          |          d|z   | |          z  |dz   ||          z  z             t          t	           ||          |          |dz    | |          z  d|z   ||          z  z             f}t           | |          t
          t          t          |dz  |                                                    z  t          t          t          |dz  |                                                    z  z   t          t          d|z  |                                                    z            t           ||          t
           t          t          |dz  |                                                    z  t          t          t          |dz  |                                                    z  z   t          t          d|z  |                                                    z            g}t          ||          d	d
d
gfk    sJ t          t	           | |          |          d|z   | |          z  |dz   ||          z  z             t          t	           ||          |          |dz    | |          z  d|z  d|dz  z  z    ||          z  z             f}t           | |          t
          t          t          d           dz  t          dd          z   t          |dz  |                                          z            z  t          t          t          d          dz  t          dd          z   t          |dz  |                                          z            z  z   t          t          d|z  |                                                    z            t           ||          t
          t          d           dz  t          dd          z   z  t          t          d           dz  t          dd          z   t          |dz  |                                          z            z  t          t          d          dz  t          dd          z   z  t          t          d          dz  t          dd          z   t          |dz  |                                          z            z  z   t          t          d|z  |                                                    z            g}t          ||          d	d
d
gfk    sJ t          t	           | |          ||          d | |          z  d ||          z  z             t          t	           ||          ||           | |          d ||          z  z             f}t          t          d           dz              }t          t          d           dz             }t          t          d          dz              }t          t          d          dz             }	t           | |          dt
          z  t          ||z            z  dt          z  t          ||z            z  z   dt          z  t          ||z            z  z   dt           z  t          ||	z            z  z             t           ||          t
          |dz  dz
  z  t          ||z            z  t          |dz  dz
  z  t          ||z            z  z   t          |dz  dz
  z  t          ||z            z  z   t           |	dz  dz
  z  t          ||	z            z  z             g}t          ||          d	d
d
gfk    sJ t          t	           | |          ||          d | |          z  d ||          z  z   dz             t          t	           ||          ||          d | |          z  d ||          z  z   dz             f}t          t          d           dz  t          dd          z              }t          t          d           dz  t          dd          z             }t          t          d          dz  t          dd          z              }t          t          d          dz  t          dd          z             }	t           | |          dt
          z  t          ||z            z  dt          z  t          ||z            z  z   dt          z  t          ||z            z  z   dt           z  t          ||	z            z  z   t          dd           z
            t           ||          t
          |dz  dz
  z  t          ||z            z  t          |dz  dz
  z  t          ||z            z  z   t          |dz  dz
  z  t          ||z            z  z   t           |	dz  dz
  z  t          ||	z            z  z   t          d!d           z             g}t          ||          d	d
d
gfk    sJ t          t	           | |          ||          dt	           ||          |          z  z
  d | |          z  z   d
          t          t	           ||          ||          dt	           | |          |          z  z   d ||          z  z   d
          f}t           | |          t
          t          |t          dd          t          d          dz  z   z            z  t          t          |t          dd          t          d          dz  z   z            z  z   t          t          |t          d           dz  t          dd          z   z            z  z   t           t          |t          d           dz  t          dd          z   z            z  z             t           ||          t
           t          |t          dd          t          d          dz  z   z            z  t          t          |t          dd          t          d          dz  z   z            z  z   t          t          |t          d           dz  t          dd          z   z            z  z
  t           t          |t          d           dz  t          dd          z   z            z  z             g}t          ||          d	d
d
gfk    sJ t          t	           | |          ||          d|z  t	           ||          |          z  d ||          z  z
            t          t	           ||          ||          d|z  t	           | |          |          z  d | |          z  z
            f}t          d          dt          d"d          z  z  t          t"                    z  t%          t          d          dt          d"d          z  z  |z  dz            z  dz  t          dt          d          z  |dz  z  dz            |z  z
  }
t          d           dt          d"d          z  z  t          t"                    z  t'          t          d          dt          d"d          z  z  |z  dz            z  dz  t          d#t          d          z  |dz  z  dz            |z  z
  }t           | |          t          |z  |dt
          z  |
z  dt          z  |z  z   z  z             t           ||          t           |z  |dt          d          z  t
          z  |
z  dt          d          z  t          z  |z  z
  z  z             g}t          ||          d	d
d
gfk    sJ t          t	           | |          |          d$ | |          z            t          t	           ||          |          d% | |          z  d ||          z  z             t          t	           ||          |          d | |          z  d ||          z  z   d ||          z  z             f}t           | |          t
          t          d$|z            z            t           ||          d%t
          z  t          d$|z            z  d&z  t          t          d|z            z  z             t           ||          d't
          z  t          d$|z            z  d(z  dt          z  t          d|z            z  dz  z
  t          t          d|z            z  z             g}t          ||          d	g d)fk    sJ t          t	           | |          |          d ||          z  d* ||          z  z
            t          t	           ||          |          d ||          z  d | |          z  z
            t          t	           ||          |          d* | |          z  d ||          z  z
            f}t           | |          dt(          z  t          d+          t
          z  t          t          d+          |z            z  z   dt
          z  dz  d,t          z  dz  z   t          t          d+          |z            z  z             t           ||          d*t(          z  t          d+          t          z  t          t          d+          |z            z  z   d-t
          z  dz  dt          z  dz  z
  t          t          d+          |z            z  z             t           ||          dt(          z  t          d+          d.t
          z  dz  d*t          z  dz  z
  z  t          t          d+          |z            z  z   d*t
          z  dt          z  z
  t          t          d+          |z            z  z             g}t          ||          d	g d)fk    sJ t          dt	           | |          |          z  d/ ||           ||          z
  z            t          dt	           ||          |          z  d ||           | |          z
  z            t          dt	           ||          |          z  d | |           ||          z
  z            f}t           | |          t(          dt          d          z  t
          z  t          dt          d          z  |z            z  z   dt
          z  dz  d0t          z  dz  z   t          dt          d          z  |z            z  z             t           ||          t(          dt          d          z  t          z  t          dt          d          z  |z            z  z   d1t
          z  d2z  dt          z  dz  z
  t          dt          d          z  |z            z  z             t           ||          t(          dt          d          z  d3t
          z  d4z  d5t          z  d4z  z
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            t          t	           ||          |          d | |          z   ||          z             f}t           | |          t
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            t          t	           ||          |          d | |          z  d ||          z  z
            f}t           | |          t
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          t          d|z            z  dt          z  t          |          z  z   dt          z  t          |          z  z             g}t          ||          d	g d)fk    sJ t          t	           | |          |           | |           ||          dz  z            t          t	           ||          |           ||          dz            f}t           | |          t
          t          d6dt          z  d|z  z   z  t          d6d          z            z            t           ||          d6dt          z  d|z  z   z  t          d"d          z             t           | |          t
          t          d6d6dt          z  d|z  z   z  t          d"d          z  z            z            t           ||          d6dt          z  d|z  z   z  t          d"d          z            t           | |          t
          t          t*           d6dt          z  d|z  z   z  t          d"d          z  z            z            t           ||          t*           d6dt          z  d|z  z   z  t          d"d          z  z            t           | |          t
          t          t*          d6dt          z  d|z  z   z  t          d"d          z  z            z            t           ||          t*          d6dt          z  d|z  z   z  t          d"d          z  z            g}t          ||          d	d
d
gfk    sJ t          t	           | |          |          t          d | |          z             ||          dz  z            t          t	           ||          |           ||          dz            f}t           | |          t-          t
          dd6dt          z  d|z  z   z  t          d"d          z  z  z
             dz            t           ||          d6dt          z  d|z  z   z  t          d"d          z             t           | |          t-          t
          dd6dt          z  d|z  z   z  t          d"d          z  z  z              dz            t           ||          d6dt          z  d|z  z   z  t          d"d          z            t           | |          t-          t
          dt*          z  d6dt          z  d|z  z   z  t          d"d          z  z  z              dz            t           ||          t*           d6dt          z  d|z  z   z  t          d"d          z  z            t           | |          t-          t
          dt*          z  d6dt          z  d|z  z   z  t          d"d          z  z  z
             dz            t           ||          t*          d6dt          z  d|z  z   z  t          d"d          z  z            g}t          ||          d	d
d
gfk    sJ t           | |          |t	           | |          |          z  t	           | |          |          t	           ||          |          z  z             t           ||          |t	           ||          |          z  t	           ||          |          dz  z             f}t           | |          t
          t          z  t
          |z  z             t           ||          t          dz  t          |z  z             h}t          ||          d	d
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gfk    sJ d S )7Nzx, y, z)clstr0      ir-      Tr   r,   r/   )   i  +   '      r*         Q   :      M   /         m               r)                  )r   r   r   r+         ii      i
   ir3      )r   r   r
   r	   r   r6   r   r   r7   r   r   r   r   r   doitC3C4r   r   r   C0r   r   )r"   r#   zrC   r>   r=   root0root1root2root3I1I2s               r&   test_checksysodesolrq   M   s+   iX...GAq!sA
T!!A$$q\\1QQqTT6
"
"BtAAaDD||R!W$=$=	>BaaddAbDRQZ\***QrT#aQik2B2B-BBCCqqttRQZ]3r$q''z!|,,,qay|C$q''	!<L<L/LLMMOC"c""taVn4444
T!!A$$q\\1QQqTT6AaaddF?
+
+RQQqTT!b1g11Q44>O-P-P	QBaaddAbDQdAQ ?@AAAAbDQPTUYPZPZ[\P\ROOQ N J J E   11Q44d4jj[]Xb!__%D!Ec!dSWjj[YZ]ROON K G G "HROOd4jjl:;C4::a<RZ[]_`RaRaCa@b<c<cc"d e efC "c""taVn4444
T!!A$$q\\11Q44!!A$$;
'
'D1aLL"QQqTT'AaaddF:J)K)K	LBaaddRDGGAIaK(((2c$q''!)A+.>.>+>>AhqRSnnDT@U@UUVVqqttr!td1ggbjl"CQ	!$4$44Q
1qD9	
d1ggaik

8 (1a.. 0112 3 34C "c""taVn4444
T!!A$$q\\11Q44!!A$$;?
+
+RQQqTT!a!fq1voPR>R-S-S	TBaaddBs1tAwwhl+,,,r#aa1o2F2F/FFROO  1rDGG8a<0Qa15E1F1FFQGGN JDGGaK!!J"  "$,QNN 3 4 45C "c""taVn4444
T!!A$$q\\11Q44!!A$$;+
,
,baadd1r!!A$$w1~PR?R.S.S	TBaaddRDGGAI&CQ	NN)::CFFBXbRS__TUUqqttd1ggbjT!WWQY'$q''"*Sa^^*CCSVVKhWZ\]N^N^^__aC"c""taVn4444
T!!A$$q\\1Q3qqtt8a!f,
-
-r$qqttA,,!!A$$1QQqTT@Q/R/R	SBaaddRHQNN//11222Rhq!nn=R=R=T=T<T8U8U5UU!A#q		!!""# $ $%'!r#x1~~6K6K6M6M2N2N/NsXa^^!!###$$$0%&)8AaC+;+;*A*A*C*C&D&D/E &F &FGC "c""taVn4444
T!!A$$q\\1Q3qqtt8ad11Q44i/
0
0"T!!A$$q\\AqD51:PQRSPSTUTUVWTXTXPXCX2Y2Y	ZBaaddRXadA..4466777"S(1a4QRBSBSAYAYA[A[=\=\:\\!A#q		!!""# $ $%'!s3Aq8I8I7O7O7Q7Q3R3R/RsHQT1##%%&&&0'(+Xac1-=-=,C,C,E,E(F(F/G &H &HIC "c""taVn4444
T!!A$$q\\1Q3qqtt8ad11Q44i/
0
0"T!!A$$q\\AqD51:QRSTQTUVWXZ[W[U[Q[]^]^_`]a]aPaCa2b2b	cBaaddRd2hhYq[8Aq>>9HQT1<M<M;S;S;U;UUVVVsDHHQJ!Q'(1a4*;*;)A)A)C)CCDDDEFI8TUVWTWYZK[K[JaJaJcJcFdFde f fqqttb488)A+A./d2hhYq[8Aq>>5QT\]^`a]acdTeTeSkSkSmSm4m0n0nnR
Xa^^#$S$r((1*x1~~*EQRTUQUWXHYHYG_G_GaGa)a%b%bbcdgiqrsturuwxiyiyhh  iB  iB  eC  eCC D DEC "c""taVn4444
T!!A$$q^^QqqttVb1g-
.
.4!Qq>>11Q44!AAaDD&=0Q0Q	RB488)a-   E$r((QE$r((Q,EbAEaaddBrE#ag,,&Bs1U7||);;beC%LL>PPSUVXSXY\]^_d]dYeYeSeeffqqttRAs1U7||+b%(Q,.?AeG.LLq1c!E'll"#%'A%6s1U7||%CD E EFC "c""taVn4444
T!!A$$q^^QqqttVAaaddF]2-
.
.4!Qq>>1QQqTT6!AAaDD&=QSCS0T0T	UB499*Q,"a0111E$s))AQ/00E$s))A+Q/000Ec1xA.//EaaddAbDQuW%"S5\\(99AbDQuW<MMPQRTPTUXYZ[`Y`UaUaPaadlmprtduduuvvqqttRAs1U7||+b%(Q,.?AeG.LLq1c!E'll"#%'A%6s1U7||%CDFNsTVFWFWX Y YZC "c""taVn4444
T!!A$$q^^aQQqTT!n,q1v5a
8
8"T!!A$$q^^aPTUVUVWXUYUYZ[P\P\n=\_`ababcdaeae_e=efg:h:h	iBaaddBs1hq!nntCyy{:;<<<r#aRSUVZ^_bZcZcdeZeIeFfBgBg?ggs1tCyyjlXa^^+,---.023q499*Q,RSUV:W7X3Y3Y0YZ [ [\^_`_`ab_c_cfhehilmnpxyz|}p~p~  BF  GJ  BK  BK  LM  BM  qM  nN  jO  jO  fOQAc1,-	.	..f/13CDII:a<(STVW..;X8Y4Z4Z1Zf[]_`cdehlmphqhqgqrsgsv~  @A  CD  wE  wE  hE  eF  aG  aG  ^GfG ]H ]HIC "c""taVn4444
T!!A$$q^^QqSaadd1-a!f4
5
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q''!Xa^^#	#DHH	,SaHQNN9J1J11LQ1N-O-O	OPQ	QTWXZ[_`a[b[bXbcdfgcgXghiXiTjTjklTl	lBaaddBqD1ad2g"R/00112aaddBqD1aQiPRlSUoXYZ^_`ZaZaXabdXdegXgFgCh<h3i3i
jC"c""taVn4444
T!!A$$q\\2aadd7
#
#RQQqTT!b1ga!fn%E%Er$qqQRttTU,,XYZ[Z[\]Z^Z^X^_`ababcdaeae_eXefghihijkhlhlflXlGmGm	nBaaddBs2a4yyL!!2aaddBrE#bd))OB,>C!HH,L#M#MqqttSVC1IIc!AbDQqSM!O3bQqSkABBDC"c""tYYY&77777
T!!A$$q\\!AAaDD&AAaDD.
)
)"T!!A$$q\\!AAaDD&11Q44-*H*HDQRQRSTQUQUVWLLY[\]\]^_\`\`Y`abcdcdefcgcgagYgIhIh	iBaaddAbD499R<DIIaK(8(888BrE!Gc"fQh<NPSTXY\T]T]^_T_P`P`;``aaqqttRUT#YYr\#d3iik"2"222c"fQhBq6H#dSViiXYkJZJZ5ZZ[[qqttQrTDIIr"uQwBq01#d3iik2B2BBBbeaPRdlTWX\]`XaXabcXcTdTdEddeegC "c""tYYY&77777
QtAAaDD||^C1aaddO
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,R$qqttA,,sAAaDD1I-O-OPRSTUYZ[Z[\]Z^Z^_`UaUaSabeghghijgkgklmlmnolplpgpbqPrPr	sBaaddB4772c!DGG)A+&6&666"R%'CF2I:MsSTUYZ[U\U\S\]^S^O_O_9__``qqttR!DGG)B,s1T!WW9Q;////3r6"9r"uQw3FAdSTggIVWKHXHX2XXYYqqttR!DGG)RU2X2b01#aQik2B2BBBbeAgPRSUPUVWPWFWY\]^_cde_f_f]fgh]hYiYiEiijjlC "c""tYYY&77777
T!!A$$q\\!AAaDD&11Q44-
(
(D1aLL11Q44!!A$$qqtt9K)L)LRPTUVUVWXUYUYZ[P\P\]^_`_`ab_c_c]cdedefgdhdh]hMiMi	jBaaddBs1Q3xxK"Q$s1Q3xx-/"S1XX+=1a4AaC@PQR@RRUWXYUYZ]^_`a^aZbZbUbbeghklmnolohphpeppqqqqttRAaC[2a4AaC=(2c!A#hh;6AqDQqS9I!9KKbQRdSVWXYZWZS[S[m[\\qqttQrT#ac((]QrT!VC!HH_,r#ac(({:R1WS1XX=MMPRSTPTUXYZ[\Y\U]U]P]]`bcfghijgjckck`kkllnC "c""tYYY&77777
T!!A$$q\\!AAaDD&11Q44-!AAaDD&0
1
1"T!!A$$q\\!AAaDD&11Q44-QRSTSTUVSWSWQWBW2X2XY[\`ababcdaeaefg\h\hijklklmnkokoiopqrsrsturvrvpvivYwYw	xBaaddBs1Q3xxK"s1vvg#a&&&8"99B#a&&3q66@Q<RRSSqqttR#a&&1SVV8#$r1SVV8c!ff+<'==>>11Q44CPQRSPSHHWXY[W[\_`a\b\bWbIbefgieijmnojpjpepIp@q@qsC"c""tYYY&77777
T!!A$$q\\!!A$$qqttQw,
'
'D1aLL1q)A)A	BBaaddBsB"qs
OxA?@@@AA2aaddbRSTVRVYZ[\Y\R\o`hijlm`n`nMnLnCoCoqqttRBAbD1Q3J(1a..88999::Bqqttb!B$QRSTQT*oX`abdeXfXfEf<g<gqqttRQBAbD1Q3J(1a..88999::BqqttaRQrTTUVWTWZ[cdegh[i[iHiEi<j<jqqttRAr1R4!A#:!Q77888992aaddAr1R4RSTURU:YabcefYgYgFgDg;h;hjC "c""taVn4444
T!!A$$q\\3q1v;;qqttQw.
/
/4!Q<<1q0I0I	JBaaddSaQrTAaCZ8Aq>> AAABBB1DEEr!!A$$RTVWXZVZ]^_`]`V`RadlmnpqdrdrQrPrGsGsqqttc"q"adQqSj/HQNN:::;;;A=>>11Q44"aPRdUVWXUXj/\defhi\j\jIj@k@kqqttc"qsB"qs
Ohq!nn<<<===a?@@"QQqTTA2rSTUWSWZ[\]Z]S]aijkmnaoaoNoKoBpBpqqttc"qsB"qs
Ohq!nn<<<===a?@@"QQqTT1bRSTVRVYZ[\Y\R\o`hijlm`n`nMnKnBoBoqC "c""taVn4444
QQqTT!D1aLL.aadd1d11Q44ll!::
;
;R!QtAAaDDQR||^TXYZYZ[\Y]Y]^_T`T`bcTcEc=d=d	eBaaddBrEBqDL!!2aaddBEBqDL#9#9
:C"c""taVn444444r%   N)2sympy.core.functionr   r   r   sympy.core.numbersr   r   r   sympy.core.relationalr	   sympy.core.symbolr
   r   &sympy.functions.elementary.exponentialr   r   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   r   'sympy.functions.special.error_functionsr   r   r   sympy.integrals.integralsr   sympy.solvers.ode.subscheckr   r   sympy.functionsr   r   sympy.testing.pytestr   r   ri   r6   r7   rg   rh   ur"   r#   rj   r   r   r   r@   rq   r$   r%   r&   <module>r      s   < < < < < < < < < < 0 0 0 0 0 0 0 0 0 0 $ $ $ $ $ $ / / / / / / / / = = = = = = = = 9 9 9 9 9 9 ? ? ? ? ? ? ? ? C C C C C C C C C C . . . . . . C C C C C C C C , , , , , , , , - - - - - - - - WV__ BBWW4(((
1aHSMMHSMMHSMM 03 03 03f~5 ~5 ~5 ~5 ~5r%   