Abstract : The main goal in this work  to find the general solution for some kind of linear second order homogenous differential equations with variable coefficients which have the general form    , by using the substitution   ,which transform form the above equation to Riccati equation 

Shaaban, Rehab A. .

English

PhilPapers

International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 40 The General Solution for a linear Second Order Homogenous Differential Equations with Variable Coefficients Rehab  A .Shaaban Dep .of Mathematics .College of Education for girls. University of Kufa . Iraq   Abstract : The main goal in this work  to find the general solution for some kind of linear second order homogenous differential equations with variable coefficients which have the general form  0)()(  yxQyxPy , by using the substitutiondxxZey)(  ,which transform form the above equation to Riccati equation . Keyword :- Differential equations , Riccati equation , Bernoulli equation . 1. INTRODUCTION  Many researchers  in this field of differential equations , may face a difficult in solving the linear second order differential equations by using known methods . Therefore  ,they are trying to solve these equations by using the power series or the Frobenius method [1] . Kathem [ 2 ] gave a method for solving the above equation ,this method depends to find a function  )(xZ   such that dxxZey)(. Kathem [ 2]   only gave examples which enable to find the general soluation by using this substitution .   2- Bernoulli Equation [3]          The general form that of  Bernoulli equation  has is written as   1)()(  nnyxqyxpy where  p  and  q   are functions of, x   (or constants) 3-  Riccati Equation [4]  The general form of Riccati equation is written as   )1...(2)()()( yxhyxgxfy   where  gf ,   and  h  are given functions of  x   ( or constants) . We can solve it , if one or more particular solutions of  )1(  can be found by inspection or otherwise . The general solution of   )1(  is easy to be obtained by the following conditions i- If 1y  is a known particular solution , then the general solution can be obtained by  the assumption :-  1yyU    , then )1(  transformed into  Bernoulli equation 2)12()( hUUhygxU   , so ,  the general solution of   )1(   is given by   dxhygexxdxxxhcyy)12()(;)()()()1(    International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 41 ii-If 1y  and  2y  are two known  particular solutions  , then  the general solution of  )1( can be found  by the  assumption  1)2( yyyyU   , then the general solution is given by :- dxyyxheyyCyy)21()()2(1    , where C  is an arbitrary constant .  iii- If 21 , yy   and  3y  ,are three known  particular solutions, say   then the general solution of equation  )1(  is given as : Cyyyyyyyy)()()()(113231   , where C  is an arbitrary constant   4-  How  Find The General Solution for the Linear Second Order Differential Equations            We can solve the equation                          )2...(0)()(  yxQyxPy    by the following cases: i- If )(xP  and )(xQ are constants say axP )(  and bxQ )(  then the equation )2(  becomes )3...(0)()()( 2  bxaZxZxZ   , and the solution of )3( is given by :- a)                     xabCxabCxaey42sin242cos12  , if 42ab   ,where  and   are arbitrary constants  b)                                 CxxaeAy  2 , if 42ab  ,where A  and C  are arbitrary constants .  proof :-  International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 42 a) Since   0)()(2)(  bxaZxZxZ , so 422;22204222abdCxdaZdZdxabaZdZ  fACandfACwherexabCxabCxaeygeAdxfxaeAygxadxfedxdeydcfadxfdZCxdaZdsin2cos1;)42sin242cos1(2);cos(22)cos(ln2a-dx)-(ftan;2)tan(2/1tan1  4.1.Example:- For solving the differential equation  32,032  bandayyy   , since 42ab   Then , by using the above formula , we get the general solution which has the form )2221(,)2221(,3)22222221()2sin22cos1()13sin213cos1(iCCBiCCAwherexBexAeixexeCxexeCxeyixCixCxeyxCxCxey b) If 1122;1221024CCCxaZxCaZdxaZdzab   Since  International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 43   2;222ln21)(CeACxxaeAyCxaCxeydxaCxeydxxZey   4.2.Example:-For solving the differential equation 4,4;044  bayyy  we will use the general form in the above formula and we get )(2,)(2 BAxxeyACBBAxxaey   ii-If 0)( xQ  , then the general solution is given by :-   BdxpdxeAy proof:- Since 2020)()(2)( ZZPZZPZZQxZPxZxZ   , this is like Bernoulli equation , to solve it , let  tZ 1 1 Ptt  .this equation is linear , and its integrating factor is given by:-   1,)(1)(ln)()()()()()()()()()()(.CeABdxdxxpeACdxdxxpeeydxdxdxxpedxxpeeydxdxxpedxxpeZdxdxxpetdxxpedxdxxpedxdxxpexptdtdxxpedxxpeFI  4.3.Example :- For solving the differential equation 02 yxy   , we use the general form in the above formula and we get  International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 44   BdxdxxpeAy)(        BdxdxxeAy2 31;31ln2 AABxAyBdxxeAy         iii-If )(2)( xQxP   , then the equation )3( can be solved by the assumption )()( xQxZu    , since   0202202  QZZQQZZQPZZZ    ,  to solve this equation , let PQuuuPQuPQuQQuZQZu 2022  , this is Riccati equation  , with PQxkandxgxf )(0)(,1)(  Now , there are many cases  1-If 1u is a known solution to the last equation , then the general solution is given by   dxdxuedxxQueAy 12)(1 proof:-    The assumption 1uud   transforms the equation to Bernoulli equation which has the  form:- 2120122 dduddudd   ,    to solve it , we set       112221  tuttddtddtd  , this is linear equation , and its integrating factor is given by :- dxueFI 12. ,so the general solution of the last equation is given by :-  International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 45                ceAdxdxuedxxQueAyCdxxQudxdxueeydxxQudxdxuedxueeyxQudxdxuedxueZudxdxuedxuexQZudxdxuedxueudxdxuedxueuudxdxueddxuedxdxuetdxue   ;12)(1)(112ln)(11212)(1121211212)(112121212112121212  . 2-If 1u and 2u  are two known solutions , then the general solution of the last equation is given by :- tconsCdxQdxuueCdxuueCuuey tan;2112121    proof:- From Riccati equation we get                 CdxuueuuCuu ;2121  is any arbitrary constant so      dxQdxuueCdxuueCuueydxuueCdxuueCuuu21121212112121  International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 46  3-if 21 ,uu  and 3u  are three known solutions , then the general solution of the last equation is given by :- tconsCuuuuxJdxxQxCJuxCJuey tan;2313)(;)()(12)(1  proof:- From Riccati equation we get :         CuuuuCuuuu;231321  is any arbitrary constant            2313)(;)()(12)(1)()(12)(1)(12)(12)()(12313)(;)(21uuuuxJdxxQxCJuxCJueyxQxCJuxCJuZxCJuxCJuuuxJCuxJCuuuuuuxJxJCuuuu Note:-  Some of these equations can be transformed into variable separable equations and don’t need the above formula to find the general solution  4.5. Example :- For solving the differential equation 2)(,2)(;022 xxQxxPyxyxy   ,   by using the equation )3(  we get                        002 222  xZZxxZZZ        ,   let                 xCxZCxtCxtdxtdttttZtxZtanhtanhtanh01011122, since  International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 47   CaeBCaeAxBxAxeyCxaxeyaxCxeydxxCxeydxxZeysinh,cosh;sinhcosh221)cosh(221221coshln)tanh()(   iv) If )(xP  and )(xQ  are not any one of the above cases , then the equation 0)()(2  xQZxPZZ  is like Riccati equation . As a result  then there are three  cases :  1-If 1Z  is a known solution to it, then the general solution of )1( is given by :- aeAdxdxZPedxZeAy  ;121 Proof :-       The assumption uZZ  1 transforms the equation to Bernoulli equation which has the  form:-   02 21  uuZPu    , to solve it , we assume tu 1          12 1  tZPt this is a linear equation , and its integrating factor (I.F) is given by :- aeAdxdxZPedxZeAyadxZedxdxZPeeydxZdxdxZPedxZPeeyZdxdxZPedxZPeZdxdxZPedxZPetdxZPeFI          ;121112ln11212112121212.12. 2- If 1Z and 2Z  are two known solutions of it , then the general solution of this equation is given by  :- International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 48 tconsCdxdxZZeCdxZZeZCZey tan;2112121   proof:- From Riccati equation  , we can write                           dxdxZZeCdxZZeZCZeydxZZeCdxZZeZCZZdxZZeZCZdxZZeCZ211212121121212121211 3-If 21 ,ZZ and 3Z are three known solutions of it , then the general solution of this equation is given by :- 2313)(tan;)(12)(1ZZZZxJandtconsCdxxJCZxJCZey  proof:- From Riccati equation , we can write                                     CZZZZCZZZZ;231321 be any arbitrary constant                      dxxJCZxJCZeyxJCZxJCZZZxJCZxJCZZZZZZxJxJCZZZZ)(12)(1)(12)(12)()(12313)(;)(21 4.6..Example :-  For solving the differential equation  0222 yxyxy   ,   we use the general form in the above formula , which is  aeAdxdxZPedxZeAy  ;121   ,  International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X   Vol. 3 Issue 4, April – 2019, Pages: 40-49   www.ijeais.org 49 now, let xZ11  ( which is a particular solution of Riccati equation )           1;23133ln4ln41ACBBxxACxAxdxxexeAydxdxxedxxeAy REFERENCE  [1] Soloughter ,Dan [ 2000 ,Section 8.7]    http : //  math .furman .edu /~ des/book/Difference Equation to   Differential Equations . [2] Kathem , Athera Nema ., “ Solving special Kinds  of  Second  Order Differential Equations by using the substitution  dxxZey)(” , Msc ,  thesies , University of Kufa , College of Eduction , Departmant of  Mathematies , 2005 . [3]  Braun ,  M. ,“ Differential Equation and Their Application “ , 4 th ed , New York : Spering – Verlag , [1993 ]. [4] Murphy . G . M . , “ Ordinary Differential Equation and their Solutions “ , D .Van Nastr and Company , Inc . , New York , [1960] . 

The General Solution for a linear Second Order Homogenous Differential Equations with Variable Coefficients

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The General Solution for a linear Second Order Homogenous Differential Equations with Variable Coefficients

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