Previous Work                  Variation of Parameters             Conclusion
                                MATH312
                Section 4.6: Variation of Parameters
                           Prof. Jonathan Duncan
                               Walla Walla College
                            Spring Quarter, 2007
  Previous Work                   Variation of Parameters              Conclusion
  Outline
       1 Previous Work
       2 Variation of Parameters
       3 Conclusion
  Previous Work                    Variation of Parameters              Conclusion
  Why we Need Another Method
       Wenowhave a procedure for solving some linear differential
       equations with constant coefficients, but it is far from complete.
       Example
       The following differential equations can not be solved by
       annihilators and variation of parameter (why not?):
            y′′ + y = cos2 x
             2 ′′     ′   2     1      3
            x y +xy + x −            =x4
                              √4
            2y′′ + 2y′ + y = 4 x
       To solve such equations, we turn to the methods used in solving 1st
       order equations.
  Previous Work                    Variation of Parameters               Conclusion
  Variation of Parameters with 1st Order DEs
       When solving a first order non-homogeneous linear differential
       equation, we used a method called variation of parameter to find a
       particular solution y .
                           p
       Variation of Parameter
       The first order linear differential equation dy + P(x)y = f (x) had
                                                  dx
       a general solution y = yc + yp where yc is the general solution to
       the associated homogeneous equation. We found that:
                    y =e−RP(x) dx Z eR P(x) dxf(x) dx
                     p
       Note:
       The assumption with which we started is that y (x) = u(x)y (x).
                                                       p             c
       How does this generalize to 2nd order DEs?