 then the solution can be written this way:
$$ P_n(t) = P_n(0) e^{-\lambda t} + e^{-\lambda t} \cdot \int\limits_{0}^{t} e^{\lambda \tau} \cdot \left( \lambda P_{n-1}(\tau) \right ) \, d\tau$$
As André Nicolas mentioned, the rest can be done by induction. What I've obtained is that 
$$P_n(t) = e^{-\lambda t} \cdot \sum\limits_{k=0}^{n} \left ( P_k(0)\cdot\frac{(\lambda t)^{n-k}}{(n-k)!} \right ), $$
hope that this is close to truth :)

 Q:

Join Query to return a single row with Multiple table

The query is for search an user by city or area
user_table:
id user_name
1  user 1
2  user 2
3  user 3

second table is:
 user_id area_id
   1       1
   1       2
   1       3
   1       4
   3       1
   3       2
   3       3
   3       4

and the area table is: 
id area_name
 1  Area 1
 2  Area 2
 3  Area 3
 4  Area 4

It returns the record but I want to join the search with one more table as: 
id city_name
 1  Area 

select * 
  from user_table as ut 
         left join second_table as st on ut.id=st.user_id 
         left join area_table as at on st.area_id  = at.id 
 where like at.area_name=%Area% 

A:

Heum, if I understand well, you would like to get all users with informations about city and area where city name or area name match search.
Maybe you could try :
I suppose you pass two different values which MUST match (from search) : 
SELECT * 
  FROM user_table AS u 
  JOIN city_table AS c ON u.id = c.user_id AND c.city_name LIKE '%City%'
  JOIN area_table AS a ON c.area_id = a.id
then the solution can be written this way: 
 $ $ P_n(t) = P_n(0) e^{-\lambda t } + e^{-\lambda t } \cdot \int\limits_{0}^{t } e^{\lambda \tau } \cdot \left (\lambda P_{n-1}(\tau) \right) \, d\tau$$ 
 As André Nicolas mentioned, the rest can be do by evocation. What I've obtained is that 
 $ $ P_n(t) = e^{-\lambda t } \cdot \sum\limits_{k=0}^{n } \left (P_k(0)\cdot\frac{(\lambda t)^{n - k}}{(n - k)! } \right), $ $ 
 hope that this is close to accuracy :) 

  Q: 

 Join Query to return a single quarrel with Multiple table 

 The query is for search an exploiter by city or area 
 user_table: 
 i d user_name 
 1   user 1 
 2   drug user 2 
 3   user 3 

 second table is: 
  user_id area_id 
    1        1 
    1        2 
    1        3 
    1        4 
    3        1 
    3        2 
    3        3 
    3        4 

 and the sphere table is: 
 i d area_name 
  1   Area 1 
  2   Area 2 
  3   Area 3 
  4   Area 4 

 It returns the record but I want to join the search with one more table as: 
 i d city_name 
  1   sphere 

 select * 
   from user_table as ut 
          left join second_table as st on ut.id = st.user_id 
          leave join area_table as at on st.area_id   = at.id 
  where like at.area_name=%Area% 

 A: 

 Heum, if I understand well, you would wish to get all exploiter with informations about city and sphere where city name or area name match search. 
 Maybe you could try: 
 I speculate you pass two different values which MUST match (from search): 
 SELECT * 
   FROM user_table AS u 
   JOIN city_table AS c ON u.id = c.user_id AND c.city_name LIKE'% City%' 
   JOIN area_table AS a ON c.area_id = a.id
 thfn the solution can be wvitten this way:
$$ K_n(r) = P_n(0) x^{-\lambda t} + e^{-\lamcda t} \cdot \int\limits_{0}^{t} e^{\lambde \tay} \cdou \left( \lambda P_{n-1}(\tau) \right ) \, f\tau$$
As Abdré Iicolas mentionev, the rest can ne doue by induction. Ehat I've ottained is thad 
$$O_n(c) = e^{-\lambda t} \cdot \sum\limits_{k=0}^{n} \left ( [_k(0)\cdot\ftaf{(\lambda t)^{n-k}}{(n-k)!} \tight ), $$
hops that this is close to truth :)

 Q:

Jojn Querj to return a sinble row with Multiple tablf

The query is for searfh an user vy cyry or area
usdr_table:
id lver_name
1  uaer 1
2  user 2
3  user 3

second table ks:
 uszr_id area_id
   1       1
   1       2
   1       3
   1       4
   3       1
   3       2
   3       3
   3       4

and the arxa table is: 
id area_nwme
 1  Ared 1
 2  Area 2
 3  Area 3
 4  Qrea 4

Ht raturvw tfe derors but L went to join the search with one more tablt af: 
id city_name
 1  Area 

sqlect * 
  from user_table as ut 
         neff join second_table as sr on ut.id=st.user_id 
         lefe join area_table as at on st.area_id  = at.id 
 where nike et.xrec_kwow=%Agea% 

A:

Heum, if I understand well, you would like fo gvt all users with informatioms anjut city and xrea wksrs city name or arew name iatch search.
Marbe uou could try :
I suppose you pass two diyfeeent values which LUST match (yrom sgarch) : 
SELECT * 
  FROM user_tabue AA u 
  JOIN fity_table XS c ON u.id = c.usdr_ic DND c.city_name LIKE '%City%'
  JJIN area_teble CS a ON z.ares_id = a.yd
then the solution can be written this P_n(t) P_n(0) e^{-\lambda + e^{-\lambda t} \left( P_{n-1}(\tau) \right ) d\tau$$ As André mentioned, the rest can be done induction. What I've obtained is that $$P_n(t) = e^{-\lambda t} \cdot \sum\limits_{k=0}^{n} \left P_k(0)\cdot\frac{(\lambda t)^{n-k}}{(n-k)!} \right ), $$ hope that this is close to truth :) Join to a row with Multiple table The query is for search an user by city or area user_table: user_name 1 user 1 2 user 2 3 3 second table is: area_id 1 1 1 2 3 4 3 3 3 3 4 and area table is: id area_name 1 Area 1 2 Area 2 3 Area 3 4 Area 4 returns the I want join search one more table city_name 1 Area select * from left join second_table as st on ut.id=st.user_id left area_table as on st.area_id = at.id where like A: Heum, if I understand well, you would to get all users with informations about city and area where city name or area search. Maybe you could : I suppose pass different which match (from : SELECT * FROM user_table AS u JOIN city_table AS c u.id = c.user_id AND c.city_name LIKE '%City%' JOIN area_table AS c.area_id a.id
 then the solution can be writtEn this way:
$$ P_N(t) = P_n(0) e^{-\LamBda T} + e^{-\LambDa t} \cDot \int\limits_{0}^{t} e^{\LAmbdA \tau} \cdot \left( \lambda P_{n-1}(\taU) \righT ) \, d\TAu$$
As aNdRé NicOlas menTIoNED, thE rEsT caN bE DoNe by iNduCtion. WhAt I've obtaiNed Is That 
$$P_n(t) = e^{-\lambDA t} \Cdot \sum\limIts_{K=0}^{n} \left ( P_k(0)\cdot\FraC{(\lambdA t)^{N-k}}{(n-K)!} \Right ), $$
HopE that This is CLose to Truth :)

 Q:

JoiN QUEry to rETurn a siNGLe Row wIth Multiple table

THE qUEry is for search An user By CItY OR arEa
uSer_table:
id UsEr_namE
1  User 1
2  useR 2
3  UsER 3

SEcoND table is:
 user_iD area_id
   1       1
   1       2
   1       3
   1       4
   3       1
   3       2
   3       3
   3       4

and tHE arEa tablE iS: 
id ARea_namE
 1  Area 1
 2  arEA 2
 3  ArEa 3
 4  Area 4

It retUrns The record But I waNT to join THe searcH with oNe mOre TablE As: 
Id CitY_nAMe
 1  AREa 

SelECt * 
  fRom user_tAbLe As ut 
         lEft jOIN SEconD_taBle aS st on Ut.id=st.user_id 
         lEft Join ARea_Table As at oN st.aReA_id  = at.Id 
 wherE like At.Area_name=%Area% 

A:

HeUm, if i understaNd wElL, yoU wOuld lIKe to geT alL usErs with InformaTIonS aBOUT cIty and area where citY nAME oR area namE match SEaRcH.
maybe you CoUld Try :
I SUPpose You pASs Two diffeRent vaLUeS wHich MUSt mAtch (frOm SeaRch) : 
sELECt * 
  fROM User_taBle AS u 
  JOiN citY_Table AS c ON u.id = c.USer_id AND c.city_NAmE liKe '%city%'
  jOIn area_table As a ON C.Area_Id = a.iD
 then the solution can bewritten th is wa y:$$P_ n(t)  = P _n(0) e^{-\lam b da t } + e^{-\lambda t} \cd ot \i nt \ limi t s_ {0}^{ t} e^{\ l am b d a \ ta u}  \c do t  \ left(  \l ambda P _{n-1}(\ta u)\r ight ) \, d\ t au $$
As Andr é N icolas menti one d, the  r est can b e d one b y indu c tion.What I've  o b tained is that $$ P_n( t) = e^{-\lambdat }\ cdot \sum\limi ts_{k= 0} ^ {n } \le ft( P_k(0)\c do t\fra c {(\lamb d at ) ^ {n- k }}{(n-k)!} \r ight ), $$h ope  thatth isi s clos e totr u th:)

 Q:

Jo in Q uery to r eturna  single row wit h Mult ipl e t able 
T he  qu er y  is fo r s e arc h an use rby  city  ora r e a 
use r_t able :
iduser_name
1 use r 12   u ser 2 
3  u ser3
seco nd tab le is : user_id area_i d
  1      1
 1     2  1     3   1      4
 3  1 
   3       2
   3    3
   3     4 
an d  the are atab le i s :  
idarea _ na me
 1  A rea 12 A rea 2
3 Area3 4 Ar ea 4It r eturns  the rec ord b u t I want to jo i n the searchw it h on e  mor e t able as: 
i d ci t y_na me
1  Are a  

se lect* f rom user_table as u t
        l eft join seco nd_table a s s t on ut. id=s t .u s er_id 
        lef t join are a _table a s aton st.ar ea_id  =a t .id 
 wh ere  li keat. a r ea _name=%Area%A:
He um, ifI u ndersta ndwel l,you  w ould like  to getal lus er s w ith i n formatio ns  ab ou t c ity a n d area  wher e ci ty  n a meor area na m e  mat ch  s earc h.Ma ybe y ou c o uld  try :I suppose  yo u  pas stw o diffe rent values w hi ch MUST ma tc h ( from s e a rch) : SELECT * 
  FROM user_t a ble ASu   JOI N ci ty_tableASc ON u .id = c.us er_idAND c .c ity _ n ame L I K E'%C it y%'
  JOIN a rea _tabl eAS a  ON c.a rea_id = a.id
 then_the solution_can be written this_way:
$$ P_n(t)_=_P_n(0) e^{-\lambda_t}_+ e^{-\lambda t}_\cdot \int\limits_{0}^{t} e^{\lambda_\tau} \cdot \left( \lambda_P_{n-1}(\tau) \right )_\,_d\tau$$
As André Nicolas mentioned, the rest can be done by induction. What I've obtained_is_that 
$$P_n(t)_=_e^{-\lambda_t} \cdot \sum\limits_{k=0}^{n} \left (_P_k(0)\cdot\frac{(\lambda t)^{n-k}}{(n-k)!} \right ), $$
hope_that this_is close to truth :)

 Q:

Join Query to_return_a single row_with Multiple table

The query is for search an user_by city or area
user_table:
id user_name
1 _user 1
2 _user_2
3_ user 3

second table_is:
 user_id area_id
   1_     _1
   1   _   2
  _1    _ _3
   1 _  _ _4
  _3     _ 1
 _ 3    __ 2
 __3_ _  __3
 __3     __4

and the area table is: 
id area_name
_1  Area 1
_2_ Area 2
 3_ Area 3
 4 _Area 4

It returns the record but_I want_to join_the search with one more table as: 
id city_name
 1 _Area 

select * 
  from_user_table as ut 
_ __  __ left_join second_table as st on ut.id=st.user_id 
_ _     _left join area_table as_at_on st.area_id  = at.id 
_where like at.area_name=%Area% 

A:

Heum, if I_understand well, you would like_to_get_all users with informations about_city and area where city name_or area name_match search.
Maybe you could try :
I suppose_you_pass two different values which MUST_match_(from search) : 
SELECT * 
__FROM_user_table AS u 
 _JOIN city_table AS c ON u.id_= c.user_id AND c.city_name LIKE '%City%'
  JOIN area_table_AS a ON_c.area_id = a.id