(a) Use the shooting method to solve y’’ −62y’ +120y =0, where the prime denotes differentiation with respect to x, given the boundary conditions y(0)=0 and y(1)=2. Solve this equation by applying the shooting method, using trial slopes in the range−0.5:0.1:0.5. Note that the exact solution is

y=1.751302152539304×10^{−26}{exp(60x)−exp(2x)}

(b) By substituting x = 1− p in the original differential equation, show that y’’+62y’ + 120y =0, where the prime denotes differentiation with respect to p. Note that the boundary conditions of this problem are y(0) =2 and y(1) =0. Solve this equation by applying the shooting method, using trial slopes in the range 0 to −150 with a step of −30 at p=0. Note that a very good approximation to the solution is y=2exp(−60p). Compare the two answers you obtain for (a) and (b). Note that an illustrative script for the shooting method is given in Section 6.2. Also solve (a) and (b)using the ﬁnite difference method, implemented in two point. Use 10 divisions and repeat with 50 divisions. You should plot your answers and compare with a plot of the exact solution.

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