Q3. Given the second-order linear homogeneous ordinary differential equa- tion with variable coefficients dy - 2.0 - d.c

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Q3 Given The Second Order Linear Homogeneous Ordinary Differential Equa Tion With Variable Coefficients Dy 2 0 D C 1 (53.43 KiB) Viewed 37 times

Q3. Given the second-order linear homogeneous ordinary differential equa- tion with variable coefficients dy - 2.0 - d.c + m(m +1)y = 0, meR, d.x2 use y(x) = 3 Anxinth to obtain 70 P} (k)a02:4–2 + P3(k)ajak-1 + bnaath = 0, n=0 where P}(k), P3 (k) are polynomials of degree 2 to be determined. Find the roots of the polynomial equation Pj (k) = 0 and comment on the coefficients ao and a, in light of the smaller one of these two roots. Next, using the smaller root, establish an explicit expression for the general term bn, and thus derive a recurrence relation between an+2 and an. Finally, using the recurrence relation you have found above, obtain the first three terms of two linearly independent series solutions in their simplest form, one with even and one with odd powers of x. [20 marks)