Chapter 8.12, Problem 14P

(a) Given that $y_1(x)$ and $y_2(x)$ are solutions of (12.19) with $f(x)=0,$ and that $y_1(a)=0,y_2(b)=0,$ find the Green function [as in (12.11) to (12.16)] and so obtain the solution (12.20). Then find the particular solution (12.21) as discussed for (12.18) and (12.21).

(b) The method of variation of parameters is an elementary way of finding a particular solution of (12.19) when you know the solutions of the homogeneous equation. Show as follows that this method leads to the same result (12.21) as the Green function method. Start with the known solution of the homogeneous equation, say $y=c_1{y}_1+c_2{y}_2$ and allow the "constants" to be functions of $x$ to be determined so that $y$ satisfies (12.19). (The $c$’s are the "parameters" which are to be "varied" in the expression "variation of parameters".) You want to find $y^{\prime }$ and $y^{\prime \prime }$ to substitute into $(12.19).$ First find $y^{\prime }$ and set the sum of the terms involving derivatives of the $c$ ’s equal to zero. Differentiate the rest of $y^{\prime }$ again to get $y^{\prime \prime }.$ Now substitute $y,y^{\prime }$ and $y^{\prime \prime }$ into (12.19) and use the fact that $y_1$ and $y_2$ both satisfy the homogeneous equation [that is, (12.19) with $f(x)=0].$

You should have the two equations: $\begin{array}{l}{c}_1^{\prime }{y}_1+c_2^{\prime }{y}_2=0,\hfill \\ c_1^{\prime }{y}_1^{\prime }+c_2^{\prime }{y}_2^{\prime }=f(x),\hfill \end{array}$

Solve this pair of equations for $c_1^{\prime }$ and $c_2^{\prime }$ [say by determinants, and note that the denominator determinant is the Wronskian as in $(12.20)\text{ and }(12.21)].$ Write the indefinite integrals for $c_1$ and $c_2,$ and write $y=c_1{y}_1+c_2{y}_2$ to get (12.21).