s09_mthsc208_hw17

s09_mthsc208_hw17 - MTHSC 208 (Differential Equations) Dr....

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Unformatted text preview: MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 17 Due Friday March 27th, 2009 (1) For each of the following ODEs, determine whether x = 0 is an ordinary or singular point. If it is singular, determine whether it is regular or not. (Remember, first write each ODE in the form y + P (x)y + Q(x)y = 0.) (a) y + xy + (1 − x2 )y = 0 (b) y + (1/x)y + (1 − (1/x2 ))y = 0. (c) x2 y + 2xy + (cos x)y = 0. (d) x3 y + 2xy + (cos x)y = 0. (2) Consider the differential equation 3xy + y + y = 0. Since x0 = 0 is a regular singular point, there is a solution of the form ∞ an xn+r . y (x) = n=0 (a) Determine the indicial equation (solve for r) and the recursion formula. (b) Find two linearly independent generalized power series solutions (i.e., a basis for the solution space). (c) Determine the radius of convergence of each of these solutions. (Hint: First compute the radius of convergence of xP (x) and x2 Q(x) and apply Frobenius). (3) Consider the differential equation 2xy + y + xy = 0. Since x0 = 0 is a regular singular point, there is a solution of the form ∞ an xn+r . y (x) = n=0 (a) Determine the indicial equation (solve for r) and the recursion formula. (b) Find two linearly independent generalized power series solutions. (c) What is the radius of convergence of these solutions? (4) Consider the differential equation xy + 2y − xy = 0. (a) Show that x = 0 is a regular singular point. (b) Show that if a0 = 0, then r = −1 is one solution for the indicial equation. (c) For r = −1 and a0 = 0, find the recurrence relation for an+2 in terms of an . (d) Still assuming that a0 = 0, write the solution from (b) as a generalized power series. (e) Which elementary function (i.e., standard everyday function) is this solution? ...
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