University of Florida/Egm4313/s12.team8.dupre/R4.4.3

Problem Statement
Find $$\displaystyle y_{n}(x) $$ for n=4,7,11 such that: $$\displaystyle y''_{n}+ay'_{n}+by_{n}=r_{n}(x) $$       (1) for x in [.9,3] with the initial conditions found i.e, $$\displaystyle y_{n}(x_{1}), y'_{n}(x_1). $$ Plot $$\displaystyle y_{n}(x) $$  for n=4,7,11 for x in [.9,3].

Solution
Using a modified version of the Matlab code used in 4.4.1, shown here below, we can solve our coefficients for each y equation (at the specified n values): Our previously solved initial conditions are: $$\displaystyle y_{n}(x_{1})=-22.153,y'_{n}(x_{1})=-55.9732 $$ Using the above code, we can solve for the coefficients and final y equation to be as shown below: For n=4: $$\displaystyle y_n(x) = 8.9210e^x - 5.6353e^{2x} -0.125x^4 - 0.583x^3 -2.125x^2 - 4.125x^1 - 4.0625 $$     (4.4.3.1) For n=7: $$ \displaystyle y_n(x) = -615.0725e^x - 3.3884e^{2x} + 0.07143x^7 + 0.6666x^6 + 4.6x^5 + 24.375x^4 + 100.4166x^3 + 305.375x^2 + 615.375x + 617.6875 $$    (4.4.3.2) For n=11: $$ \displaystyle y_n(x) = -3301815e^x + 734.893e^{2x} + 0.04545x^{11} + 0.7x^{10} + 8.0556x^9 + \cdots $$ $$ \displaystyle 77.1875x^8 + 636.3214x^7 + 4520x^6 + 27318x^5 + 137082x^4 + 549316x^3 + 1649429x^2 + 3300389x + 3301079 $$    (4.4.3.3)