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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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 Spring '09
 Staufeneger
 Differential Equations, Equations

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