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                        i               a                                                                                                                                                               Arficho, J Appl Computat Math 2015, 4:2 
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                       A                 t                                                                                                                                                                       DOI: 10.4172/2168-9679.1000210
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                          uoJ         sci      Applied & Computational Mathematics 
                     Review Article                                                                                                                                                                                                  Open Access
                   Method for Solving Particular Solution of Linear Second Order Ordinary 
                   Differential Equations
                   Arficho D*
                   Department of mathematics, Aksum University, Aksum, Ethiopia
                                              Abstract
                                                    In this paper, we derive new method for solving particular solution of linear second order ordinary differential 
                                              equations whenever one solution of their associated homogeneous differential equations is given. Also, we construct 
                                              second solution of the associated homogeneous differential equation from this new method. Moreover, we have 
                                              general solution method of linear second order ordinary differential equations without applying the two famous 
                                              methods, undetermined coefficients method and variation of parameters method, for solving their particular solution.
                   Keywords: Reduction of order; Fundamental set                                                                           otherwise, it is non-homogeneous [3].
                   Introduction                                                                                                            Linear first order differential equations
                          A differential equation is an equation that relates an unknown                                                          The linear first order ordinary differential equation with unknown 
                   function and one or more of its derivatives of with respect to one or                                                   dependent variable y and independent variable x is defined by
                   more independent variables [1,2]. If the unknown function depends                                                                                         1
                                                                                                                                                                             ( )                                                                     (2.1)
                                                                                                                                                   axy  +=axy  gx.
                                                                                                                                                      ( )             ( )               ( )
                   only on a single independent variable, such a differential equation is                                                           01
                   ordinary. The order of an ordinary differential equation is the order of                                                       Solution of linear first order differential equations: The general 
                   the highest derivative that appears in the equation [3]. In real world,                                                 solution of the equation in equation 2.1 is given by
                   there are physical problems that are ordinary differential equations.                                                                    m()xg()x
                   Thus, we need solution methods to solve these problems. A solution of                                                                 ò axydx                                                                                    (2.2)
                                                                                                                                                                 ()
                   a differential equation in the unknown function y and the independent                                                           v=          21
                                                                                                                                                                m x
                   variable x on the interval I is a function y(x) that satisfies the differential                                                                ()
                                                                                                                                                                                    ax()
                   equation identical for all x in I [2]. A solution of a differential equation                                                   Where µ(x) = exp(ò ( 0                    )dx
                                                                                                                                                                                     ax()
                   with arbitrary parameters is called a general solution. A solution of                                                                                              1
                   a differential equation that is free of arbitrary parameters is called a                                                Fundamental set of solutions
                   particular solution [2]. A solution in which the dependent variable is                                                         A set of functions y (x), y (x), · · · , f (x) is said to be linearly 
                   expressed solely in terms of the independent variable and constants is                                                                                           1         2                   n
                   said to be an explicit solution. A relation G(x,y) is said to be an implicit                                            dependent on an interval I if there exist constants c1, c2, , cn not all 
                                                                                                                                           zero, such that  cy x+cy x++· · · cy x = 0  for every x in the 
                                                                                                                                                                             ( )           ( )                  ( )
                   solution of an ordinary differential equation on an interval I, provided                                                                              11            22                  nn
                   there exists at least one function f that satisfies the relation as well as the                                         interval. If the set of functions is not linearly dependent on the interval, 
                                                                                                                                           it is said to be linearly independent. Any set y (x), y (x), · · · , f (x) of n 
                   DE on I [2]. Moreover, solutions of differential equations are classified                                                                                                                         1        2                n
                   as trivial and non-trivial solutions, general and particular solutions and                                              linearly independent solutions of the homogeneous linear nth-order 
                   explicit and implicit solutions. The general solution of linear second                                                  differential equation  on an interval I is said to be a fundamental set of 
                   order ordinary differential equations consists of solutions of their                                                    solutions on the interval.
                   associated homogeneous differential equations and their particular                                                      General solution of linear higher order differential equations
                   solutions. There are two famous methods, undetermined coefficients                                                             The associated homogeneous differential equation of 
                   method and variation of parameters method, for solving particular                                                       nonhomogeneous linear nth order differential equation
                   solutions of linear second order ordinary differential equations. Finally, 
                   one can solve for particular solution of linear second order ordinary                                                                                     11nn-
                                                                                                                                                                             ( )                            (   )                ( )
                                                                                                                                                   axy  +axy  ++· · ·  a xy   +axy = gx
                                                                                                                                                      ( )             ( )                            ( )                  ( )               ( )
                   differential equations without applying undetermined coefficients                                                                01 nn-1
                   method and variation of parameters method.                                                                                     is
                   Linear Higher Order Differential Equations
                          A differential equation
                                 nn-11                                                                                                      *Corresponding author: Arficho D, Department of mathematics, Aksum University, 
                                ( )    (   )            ( )
                           Fy(     ,  y     ,  · · · ,  y ,  y)  =  gx
                                                                        ( )                                                                 Aksum, Ethiopia, Tel: 347753645; E-mail: daniel.arficho@yahoo.com 
                                                                                                                       (n)    (n−1)         Received March 10, 2015; Accepted March 25, 2015; Published April 10, 2015
                          is said to be linear if F is a linear function of the variables y                               , y      , 
                            (1)
                   · · · , y   , y [3]. An nth-order linear differential equation in a dependent                                            Citation:  Arficho  D  (2015)  Method  for  Solving  Particular  Solution  of  Linear 
                   variable y and independent variable x defined on an interval I ÌR has                                                    Second Order Ordinary Differential  Equations.  J  Appl  Computat  Math  4:  210. 
                   the form                                                                                                                 doi:10.4172/2168-9679.1000210
                                                     1 nn-1                                                                                 Copyright: © 2015 Arficho D. This is an open-access article distributed under the 
                                                     ( )                           (   )                ( )
                          axy + axy ++ · · ·                           a xy  + axy = gx.
                              ( )             ( )                           ( )                  ( )                ( )
                            01 nn-1                                                                                                         terms of the Creative Commons Attribution License, which permits unrestricted 
                          If g(x) = 0 for all x in I, then the differential equation is homogeneous,                                        use, distribution, and reproduction in any medium, provided the original author and 
                                                                                                                                            source are credited.
                      J Appl Computat Math
                      ISSN: 2168-9679 JACM, an open access journal                                                                                                                                                 Volume 4 • Issue 2 • 1000210
                       Citation: Arficho D (2015) Method for Solving Particular Solution of Linear Second Order Ordinary Differential Equations. J Appl Computat Math 4: 
                                         210. doi:10.4172/2168-9679.1000210
                                                                                                                                                                                                                                                                                  Page 2 of 3
                                                              11nn-
                                                             ( )                                  (   )                    ( )             [2].                            2.4, we have
                               axy  +axy  ++· · ·  a xy   +axy = 0  
                                   ( )               ( )                                  ( )                      ( )
                                 01 nn-1
                                                                                                                                                                                                         (1)                 (2)                                                                                                     (2.6)
                                                                                                                                                                           a()xy+a()xy +a()xy =g(x)
                              Theorem 4.1. Let y , y , · · · , y  be linearly independent solutions of                                                                        01pp2p
                                                                  1     2             n
                       the homogeneous linear nth order differential equation                                                                                              From equation in equation 2.6 It follows that                               
                                                              11nn-
                                                             ( )                                  (   )                    ( )                    (2.3)
                               axy  +axy  ++· · ·  a xy   +axy = 0                                                                                                                                                                  (1)        (1)                                  (2)
                                   ( )               ( )                                  ( )                      ( )
                                 01 nn-1                                                                                                                                    a()()xuxy()x+a()x[()uxy +u y]+a()x[()uxy        (2.7)
                                                                                                                                                                              0 11 1 12 1
                                                                                                                                                                                     (1)   (1)        (2)
                              on an interval I. Then the general solution of the equation in 2.3 on                                                                        +2y u +=u y] gx()
                                                                                                                                                                                    11
                       the interval I is                                                                                                                                   Using equation in equation 2.5, the equation in equation 2.7 is 
                               y = cy  +cy  ++· · ·  c y   +cy, y = cy  +cy  ++· · ·  c y   +cy,                                                                   reduced to
                                        11          22                    n--1n1          nn             11        22                 n--1n1         nn
                                                                                                                                                                                         1                                     12
                                                                                                                                                                                        ( )                             (1)    ( )         ( )
                                                                                                                                                                                     éù
                                                                                                                                                                            axuy + ax[2 y u += uy]   gx  
                                                                                                                                                                               ( )                        ( )                                                ( )                               (2.8)
                                                                                                                                                                             1121                                                               1
                       where c, (i=1, 2, · · · , n − 1, n) are arbitrary constants [1].                                                                                              ê        ú                      ( )
                                     i                                                                                                                                               ëû
                              Theorem 4.2. Let y , y , · · · , y  be linearly independent solutions of                                                                     From equation in equation 2.8 It follows that
                                                                  1     2             n
                       the associated homogeneous linear nth order differential equation on an                                                                                                                            12
                                                                                                                                                                                                                 (1)     ( )                         ( )
                                                                                                                                                                                                                      ùé
                                                                                                                                                                           []axy  +2 a xy                              u  +axyu = gx                                             (2.9)
                                                                                                                                                                                ( )                   ( )                               ( )                       ( )
                       interval I, and let y  be particular solution of linear nth order differential                                                                         1121                                                    21
                                                                                                                                                                                                              ( )     ú            ê
                       equation                          p                                                                                                                                                            û            ë
                                                                                                                                                                                    (1)
                                                                                                                                                                           Let u =V. Then from equation in equation 2.9, we have
                              Then the general solution of the equation                                                                                                                                                                            1
                                                                                                                                                                                                                 (1)                              ( )
                                                                                                                                                                           []axy  +2 a xy                              v  +=a xyv  gx 
                                                                                                                                                                                ( )                    ( )            ]         [     ( )                      ( )                                   (2.10)
                                                                                                                                                                              112 1                                                 21
                                                                                                                                                                                                               ( )
                                                              11nn-
                                                             ( )                                  (   )                    ( )
                               axy  +axy  ++· · ·  a xy   +axy = X
                                   ( )               ( )                                  ( )                      ( )
                                 01 nn-1                                                                                                                                   The equation in equation 2.10 is linear first order ordinary 
                              on the interval I is                                                                                                                 differential equation. Thus,
                               y = cy  +cy  ++· · ·  c y + cy  +y,
                                          1 1           2  2                      n--1  n1          nn p                                                                   using the formula in equation 2.2, we get
                       where ci, (i = 1, 2, · · · , n − 1, n) are arbitrary constants [2].                                                                                             m()xg()x
                              General solution of linear second order differential equations:                                                                                      ò a ()xy dx                                                                                                                                 (2.11)      
                                                                                                                                                                           v=              21
                       The second order linear ordinary differential equation with unknown                                                                                                 m()x
                       dependent variable y and independent variable x is defined by                                                                                                                                                     (1)
                                                                                                                                                                                                           a()xy+2a()xy
                                                                                                                                                                   where μ(x) =                             1121
                                                                                                                                                                                             exp(ò (                                         )dx)
                                                                12
                                                               ( )                   ( )                                                                                                                             a()xy
                                  a xy  ++axy  axy = gx.                                                                                                                                                               21
                                     ( )               ( )                   ( )                  ( )                                                 (2.4)
                                   012
                              Now, to solve nonhomogeneous second order linear differential                                                                                Thus, u =  ò vdx , where
                       equations, first we solve their associated homogeneous differential                                                                                             m()xg()x                                                                                   (1)
                                                                                                                                                                                                                                                   a()xy+2a()xy
                       equations. Since the general solution of the equation in equation                                                                                                               dx                                            1121
                                                                                                                                                                                   ò a()xy                      and  μ(x) = exp(ò (                                                  )dx)
                                                                                                                                                                                           21                                                                 a()xy
                       2.4 consists of its particular solution, we search methods for solving                                                                              v=              m()x                                                                 21
                       the particular solution of equation in equation 2.4. Most authors 
                       of differential equations books used two famous methods, namely,                                                                                    Clearly, 
                       undetermined coefficients methods and variation of parameters                                                                                                      2               ax()
                                                                                                                                                                           m(x)=y exp( ( 1                         )dx)  Therefore, articular solution y (x) = u(x)
                       method to find particular solution of nonhomogeneous linear second                                                                                                 1         ò ax()                                                                         p
                       order differential equations. Here we would like to introduce new                                                                                                                    2
                                                                                                                                                                   y (x) of the equation in equation 2.4 is
                       method for finding particular solution of nonhomogeneous linear                                                                               1
                                                                                                                                                                            yx  = yx                      vdx                                                                                                                           (2.12)
                       second order differential equations. Moreover, we construct a second                                                                                   p ( )            1( )ò
                       solution of the associated homogeneous linear second order differential 
                       equation so that the set consisting of y1and y2 is linearly independent on                                                                          where
                       I from this new method.                                                                                                                                         m()xg()x
                                                                                                                                                                                   ò                   dx                              2              ax()
                              Procedure for deriving new method for particular solution of                                                                                              a ()xy                 and  m(x)= y exp(                    ( 1        )dx) .
                                                                                                                                                                           v=              21                                         1         ò ax()
                       nonhomogeneous linear second order differential equations: Let y1                                                                                                   m()x                                                         2
                       be a non-zero known solution of the associated homogeneous linear 
                       second order differential equation of the equation in equation 2.4.                                                                         Result and Discussion
                       Thus, we have                                                                                                                                       There are two particular solution methods. One is method of 
                                                           (1)                (2)                                                                                                         (2.5)
                               a()xy++a()xy                       a()xy =0                                                                                         undetermined coefficients. The general method of undetermined 
                                 011121
                              Then we assume that y (x) = u(x)y (x) as particular solution of                                                                      coefficients is limited to differential equations of the form         
                                                                           p                       1
                                                                                                                                                                                                       12
                       the equation in equation 2.4 to construct particular solution of the                                                                                                           ( )                 ( )                                              and a (x) are 
                                                                                                                                                                           axy  ++axy  axy = gx, where ax, ax
                                                                                                                                                                               ( )             ( )                 ( )               ( )                  ( )        ( )
                       equation in equation 2.4.                                                                                                                             0 1 2                                                                       01 2
                                                                                                                                                                   constant functions and g(x) is a constant k, a polynomial function, 
                              It follows that                                                                                                                      an exponential function, a sine or cosine function, or finite sums 
                                                                                                                                                                   and product of these functions [2]. Thus, this method is not general 
                                   (1)             (1)                                (1)                                                                          method for particular solution. The second particular solution method 
                               y ()x=+u(x)y(x) ux()y ()x
                                                                                                and
                                  p                            11                                                                                                  is variation of parameters method. Unlike method of undetermined 
                                   (2)             (2)                         (1)         (1)                     (2)
                               y ()x=+u (x)y(x) 2u ()xy ()x+u()xy ()x                                                                                              coefficients, this method is general method for particular solution 
                                  p                            111
                              Since y (x)=u(x)y (x) is particular solution of the equation in                                                                      because it works even if the conditions given for method of 
                       equation             p                   1                                                                                                  undetermined coefficients fail.
                         J Appl Computat Math                                                                                                                                                                                                          Volume 4 • Issue 2 • 1000210
                         ISSN: 2168-9679 JACM, an open access journal 
             Citation: Arficho D (2015) Method for Solving Particular Solution of Linear Second Order Ordinary Differential Equations. J Appl Computat Math 4: 
                       210. doi:10.4172/2168-9679.1000210
                                                                                                                                                           Page 3 of 3
                 Now we derived the third method for particular solution of                 solution of the associated homogeneous equation of the equation in 
                                (1)        (2)       as   y (x)=y (x)    vdx ,   where      equation 2.4 inserting g(x)=0 in this particular solution method. This 
                  a()xy++a()xy       a()xy =g()x            p      1   ò
                   012                                                                      implies that 
                    m()xg()x                                                                         ò 0dx . Thus choose   v= 1
                 ò a xydx  and  y =+ cy  cy  +y,                                                 v=                             m()x
                       ()                   11      22      p                                         m x
             v=       21                                                                               ()
                      m x
                        ()                                                                      Therefore,
                 Unlike variation of parameters method, we need only one 
             solution of the associated homogeneous differential equation of a                   y (x)  = y (x)   vdx ,
                                                                                                  21
             differential equation to derive particular solution of a differential                              ò
             equation. Moreover, this new method for particular solution is good                Where
             method because it works even if the conditions given for method of                       1                 2        ax()
                                                                                                 v=        and m(x)=y exp( ( 1        )dx)  is the second solution of 
             undetermined coefficients fail.                                                         m()x               1     ò ax()
                                                                                                                                  2
             Conclusion                                                                     the associated homogeneous  equation of the equation in equation 2.4. 
                                                                                            Moreover, the general solution of the equation in equation 2.4 is
                 Now, we use equations in equation 2.12 to draw conclusion about                 y =+ cy  cy  +y,
             particular solution method of the equation in equation 2.4 if y1(x)                       11      22      p
             is known solution of the associated homogeneous equation of the                where y , y  and y  are as given above. Here c, (i=1, 2) are arbitrary 
                                                                                                     1  2       p                             i
             equation in equation 2.4. Therefore, the particular solution method of         constants. 
             the equation in equation 2.4 is                                                References
                  y  x   = y x  vdx,                                                        1. Levermore D (2012) Higher-Order Linear Ordinary Differential Equations I:, 
                   p ( )    1( ) ò                                                             Department of Mathematics, University of Maryland. 
                               m()xg()x                                                     2. Dennis G, Zill (2013) A First Course in Differential Equations, Ricard Stratton, 
                             ò          dx                2        ax()
                                a xy                                1                          Los Angeles, United States of America, (10thedn).
                 Where            ()          and  m(x)= y exp(   (     )dx)
                         v=      21                       1     ò ax()
                                  m x                               2
                                   ()                                                       3. Yüksel  S  (2014)  differential  equations  for  engineering  science,  Queen’s 
                                                                                               University, Canada.
                 From this particular solution method, one can construct the second 
              J Appl Computat Math                                                                                                         Volume 4 • Issue 2 • 1000210
              ISSN: 2168-9679 JACM, an open access journal