Method of Variation of Parameters  (continued)

 

 

 

Strategy:  Substitute    yp(x),   dyp(x)/dx ,    d2yp(x)/dx2     into

 

                               d2y/dx2  +  b dy/dx  +  cy  =  f(x)

 

The result is a second equation to determine    du1/dx    and    du2/dx .       

 

 

This second equation has the form:

 

                  

           du1(x)/dx  dy1(x)/dx  +  du2(x)/dx dy2(x)/dx  =  f(x)              (equation 2)

 

 

where we used that  both    y1(x)  and    y2(x)      satisfy the homogeneous d.e.

 

                            a d2y/dx2  +  b dy/dx  +  cy  =  0

 

 

The next step is to solve equations (1) and (2), using algebra, for  du1(x)/dx  and

du2(x)/dx  .  Note, you have two equations in two unknowns.   The unknowns are

the   du1(x)/ dx  and   du2(x) / dx .     

 

 

They equations are as follows:          (repeated for you here)           

 

                          du1(x)/dx  y1(x)         +  du2(x)/dx y2(x)  =  0               (equation 1)

 

                          du1(x)/dx  dy1(x)/dx  +  du2(x)/dx dy2(x)/dx  =  f(x)    (equation 2)

 

 

 

 

 

Strategy:  Integrate expressions for  du1(x)/dx   and for   du2(x)/dx  to obtain

 

                                                   u1(x) and u2(x)    .

 

It turns out that the constants of integration end up being absorbed in the complementary solution.

 

This integration may not be simple.

 

Finally, the particular solution is             yp (x)  =   u1(x) y1(x)  +  u2(x) y2(x)

 

Click here for an example.

 



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