Consider the differential equation: xy + (1-x)y' + py = 0 for x > 0. Here...

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Consider the differential equation: xy" + (1-x)y' + py = 0 for x > 0. Here p > 0 is a real number. (1a) Show that the point x = 0 is a regular singular point of the above differential equation. (1b) Write down the indicial equation associated with the above differential equation and find its root(s). (1c) Derive an expression for the guaranteed Frobenius series solution to the above differential equation. (1d) Deduce that the expression for the solution obtained in Question (1c) reduces to a polynomial when p is a positive integer. (le) Write down the polynomial solutions of the above differential equation when p = 1, when p = 2 and when p = 3. Consider the differential equation: xy" + (1-x)y' + py = 0 for x > 0. Here p > 0 is a real number. (1a) Show that the point x = 0 is a regular singular point of the above differential equation. (1b) Write down the indicial equation associated with the above differential equation and find its root(s). (1c) Derive an expression for the guaranteed Frobenius series solution to the above differential equation. (1d) Deduce that the expression for the solution obtained in Question (1c) reduces to a polynomial when p is a positive integer. (le) Write down the polynomial solutions of the above differential equation when p = 1, when p = 2 and when p = 3.

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