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**Extra resources for Computer Algebra Recipes for Mathematical Physics**

**Sample text**

If n = 1 and f (x) = −x2 for 0 ≤ x ≤ 1/2 and f (x) = −(1 − x)2 for 1/2 ≤ x ≤ 1, use this Green function to explicitly determine y(x) and plot the result. The solution is as follows. First let’s note that x y + y = (x y ) , so that the ODE is a nonhomogeneous S-L equation with p(x) = x and q(x) = n2 /x. Now the ODE for the Green function is entered. 2. LINEAR ODES WITH VARIABLE COEFFICIENTS > 49 ode:=x*diff(G(x),x,x)+diff(G(x),x)-nˆ2*G(x)/x=Dirac(x-z); n2 G(x) d2 d G(x)) + ( G(x)) − = Dirac(x − z) 2 dx dx x The general solution of the ODE is obtained, assuming that x < z.

14) x y + y − n2 /x = f (x), where n is a positive (non-zero) integer and the boundary conditions are that y(0) must remain ﬁnite and y(1) = 0. If n = 1 and f (x) = −x2 for 0 ≤ x ≤ 1/2 and f (x) = −(1 − x)2 for 1/2 ≤ x ≤ 1, use this Green function to explicitly determine y(x) and plot the result. The solution is as follows. First let’s note that x y + y = (x y ) , so that the ODE is a nonhomogeneous S-L equation with p(x) = x and q(x) = n2 /x. Now the ODE for the Green function is entered. 2. LINEAR ODES WITH VARIABLE COEFFICIENTS > 49 ode:=x*diff(G(x),x,x)+diff(G(x),x)-nˆ2*G(x)/x=Dirac(x-z); n2 G(x) d2 d G(x)) + ( G(x)) − = Dirac(x − z) 2 dx dx x The general solution of the ODE is obtained, assuming that x < z.

04506702177 48 CHAPTER 1. 045. A bizarre feature that occurs for odd values of n is that the probability of ﬁnding the particle at x = 0 (the center of the potential well) is 0. You are referred to Griﬃths for a more thorough discussion of the quantum oscillator. In that reference, the probability distribution is drawn for n = 100. It is a trivial task to change n in the above recipe and replot the probability distribution. Try it! 4 Going Green, the Mathematician’s Way Colorless green ideas sleep furiously.