Consider the following differential equation, xy -y + 9xy = 0. Note: For each part below...

  

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Consider the following differential equation, xy" -y + 9xy = 0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals. (a) The above differential equation has a singular point at x = 0. If the singular point at x = 0 is a regular singular point, then a power series for the solution y(x) can be found using the Frobenius method. Show that x = 0 is a regular singular point by calculating: xp(x) = x² g(x) = Since both of these functions are analytic at x = 0 the singular point is regular. (b) Enter the indicial equation, in terms of r, by filling in the blank below. 0 (c) Enter the roots to the indicial equation below. You must enter the roots in the order of smallest to largest, separated by a comma. (d) You must now calculate the solution for the largest of the two indicial roots. First, enter the corresponding recurrence relation below, as an equation. Note 1: You must include an equals sign. Note 2: You must use the symbol m as your index. Note 3: am is entered as a (m), am+1 as a (m+1), etc. (e) Hence enter the first three non-zero terms of the solution corresponding to the largest indicial root. Note: The syntax for ao and a1 is a0 and al, respectively. y = Consider the following differential equation, xy" -y + 9xy = 0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals. (a) The above differential equation has a singular point at x = 0. If the singular point at x = 0 is a regular singular point, then a power series for the solution y(x) can be found using the Frobenius method. Show that x = 0 is a regular singular point by calculating: xp(x) = x² g(x) = Since both of these functions are analytic at x = 0 the singular point is regular. (b) Enter the indicial equation, in terms of r, by filling in the blank below. 0 (c) Enter the roots to the indicial equation below. You must enter the roots in the order of smallest to largest, separated by a comma. (d) You must now calculate the solution for the largest of the two indicial roots. First, enter the corresponding recurrence relation below, as an equation. Note 1: You must include an equals sign. Note 2: You must use the symbol m as your index. Note 3: am is entered as a (m), am+1 as a (m+1), etc. (e) Hence enter the first three non-zero terms of the solution corresponding to the largest indicial root. Note: The syntax for ao and a1 is a0 and al, respectively. y =