Consider the ordinary differential equation 2.22d2y dx2 dy +3x44 - (1 – 5x + 9x?)y=0. da (a) Show that this equation has

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Consider The Ordinary Differential Equation 2 22d2y Dx2 Dy 3x44 1 5x 9x Y 0 Da A Show That This Equation Has 1 (61.85 KiB) Viewed 58 times

Consider the ordinary differential equation 2.22d2y dx2 dy +3x44 - (1 – 5x + 9x?)y=0. da (a) Show that this equation has a regular singular point at x = 0. {3} (b) Find the indicial equation for a Frobenius series solution of this equation, expanding around the point x = 0. Show that one of its roots is != -1 and find the other root, which is not an integer. {8} (c) Show that for the root \ = -1, the first two coefficients in the series solution must obey a1 = 500 {5}. (d) Show that the recurrence relation for this value of l is as+2 9a, – 5a3+1 2s2 + 5s + 2 For what values of s does this recurrence relation apply? {6} (e) Hence find the solution of the ODE corresponding to 1 = -1, showing the first 3 (non-zero) terms in the series expansion. {3}