Module-
   1.Ordinary
   Differential
    Equation
  Dr. Chandra
    Shekher
     Singh                   Module-1.Ordinary Differential Equation
 Variation of
 Parameters
                                             Dr. Chandra Shekher Singh
                                              Department of Basic Science UPTTI,
                                                       Kanpur-208001, India
                                       Branch-TT,TE , Apr. 05, 2020-SEM-II
                                 Dr. Chandra Shekher Singh     Module-1.Ordinary Differential Equation
                  Outline of the Presentation
    Module-
   1.Ordinary
   Differential
    Equation
  Dr. Chandra
    Shekher             • Variation of Parameters
     Singh
                                                                            d2y            dy
 Variation of           • Working rule for Solving                             2 +P            +Qy=R
 Parameters                                                                 dx             dx
                           by Variation of Parameters, where P,Q and
                           Rare functions of x or Constants.
                        • Based Examples
                        • Based Questions
                                 Dr. Chandra Shekher Singh     Module-1.Ordinary Differential Equation
                  Variation of Parameters
    Module-        Definition
   1.Ordinary
   Differential
    Equation       The Wornskian of n functions y (x),y (x),...,y (x) is denoted
                                                                  1        2             n
  Dr. Chandra      by W(y ,y ,...,y ) and is defined to be the dterminant
                               1   2         n
    Shekher
     Singh                                                                                                 
                                                                      ′           ′                  ′     
                                                                 1 y (x) y (x) ...                  yn(x)
                                                                      1          2                         
 Variation of                                                         ′           ′                  ′     
 Parameters                                                      1 y (x) y (x) ...                  yn(x)
                      W(y ,y ,...,y ) = W(x) =                        1          2                          .
                             1   2         n                    . . . . . . . . . . . . . . . . . . . . . . . . . . . .
                                                                      ′           ′                  ′     
                                                                                                           
                                                                 1 y (x) y (x) ...                  yn(x)
                                                                       1          2
                   Examples
                       1 (Example-1) Consider the two function f (x) = x3 and
                                                                                      1
                                         2
                          f (x) = x , find Wronskian i.e, W(f ,f ) (The solution is
                            2                                                   1    2
                          given below ).
                       2 (Example-2) Consider the two function f (x) = sin(x) and
                                                                                      1
                          f (x) = cos(x), find Wronskian i.e, W(f ,f ) (The
                            2                                                         1    2
                               Dr. Chandra Shekher Singh     Module-1.Ordinary Differential Equation
                          solution is given below ) .
                    Variation of Parameters
     Module-          Definition
   1.Ordinary
   Differential        Variation of Parameters: Variation of Parameters is a method
    Equation
  Dr. Chandra         for producing a particular solution to an nonhomogeneous
     Shekher          equation by exploiting the (Usually much simpler to find)
      Singh
                      solutions to the associated homogeneous equation.
  Variation of
  Parameters
                      Working Procedure for solving
                                                     d2y +Pdy +Qy =R
                                                     dx2           dx
                      by Variation of Parameters, where P,Q and R are functions of
                      x or Constants.
                      Step-1: Re-write the given equation as
                                                    y +Py +Qy=R
                                                      2          1
                                                                                             (1)
                                    Dr. Chandra Shekher Singh         Module-1.Ordinary Differential Equation