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flyingsquirrel · #7 May 1st, 2008, 01:43 AM

If I've understood what is the "Green function", you have to begin by solving $\left\{\begin{array}{ll}\frac{\mathrm{d}^2y}{\mathrm{d}x^2}-y=0 & (1)\\y(0)=y'(0)=0 & (2)\end{array}\right.$

For this, write the characteristic equation* and find its roots. Let's call them $r_1$ and $r_2$ . In the case of two different real roots, the solutions of (1) are given by $y(x)=A\exp(r_1x)+B\exp(r_2x)$ . Then, use (2) to find the values of $A$ and $B$ . Once you have done this, will go through the variation of parameters method.

*For a differential equation which as the form $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+ a_1\frac{\mathrm{d}y}{\mathrm{d}x}+ a_2y=0$ , the characteristic equation is $r^2+a_1r+a_2=0$ .