Particular Solutions of Nonhomogeneous D.E.’s          (continued)

 5 For simple functions, f(x) one can find particular solutions  yp of  nonhomogeneous, linear differential equations with constant coefficients using the method of undetermined coefficients.  Consider the following d.e.                          y ’’ + A y ’ + B y  =  f(x)   First Step:  Take the derivative of f(x) up to the highest order in the d.e.  In the general case above it is a second order d.e.   Then the particular solution is a linear combination of the different functions obtained from the right hand side of the d.e..  i.e.  If the right hand side happened to be  23 sin 2x,  then the assumed form of the particular solution would be  yp  =   A sin 2x  +  B cos 2x  since both sin 2x  and cos 2x  appear when taking the function, f(x), and its first and second derivatives.  The assumption here is that the complementary solution to the homogeneous d.e. does not contain either sin 2x  nor  cos 2x.   The cases below summarize the simple cases for f(x) for which one can apply the method of undetermined coefficients to find the particular solution of nonhomogeneous d.e.’s.   Case 1:          f(x) = polynomial  =  a1x m + a2  x m -1 + . . . . + an                                  yp (x)  =    Amx m + Am-1  x m -1 + . . . . + Ao   Case 2:          f(x)  =  a cos kx  +  b sin kx                                  yp (x)  =    A cos kx  +  B sin kx   Case 3:        f(x)  =  a e kx  +  b e –kx                                               yp (x)  =  A e kx  +  B e –kx   Case 4:   Note -   f(x)  could involve products of the functions in Cases 1 – 3   NOTE:  If any of the f(x) appear in the complementary solution, then you need to pick             a new function in your particular solution that is linearly independent.   i.e.   Click here for an example using the method of undetermined coefficients.