ASSIGNMENT 1: SERIES SOLUTION TO ODES.
DUE: MONDAY MAY 28, IN CLASS
Problem 1: Find the general solutions of the following ODEs:
(a) 6y00 +y0 2y = 0:
(b) y00 3y0 + 2y = 4ex:
(c) x2y00 2xy0 4y = 0:
(d) 9x2y00 + 3xy0 +y = 0:
(e) x2y00 + 3xy0 + 2y = 0, with y(1) = 1;y0(1) = 2:
Problem 2: Find power series solutions about x0 = 0 of the following ODEs:
(a) y00 +xy0 +y = 0:
(b) (x2 + 1)y00 + 72xy0 +y = 0:
Problem 3: Determine the power series solution of the following initial value problems:
(a) y00 xy0 y = 0 with y(0) = 1;y0(0) = 0.
(b) (x 1)y00 (x 3)y0 y = 0 with y(0) = 3;y0(0) = 3.
Problem 4: Consider the ODE:
y00 2xy0 + y = 0:
where is a constant.
(a) Using the series expansion about x0 = 0, determine two independent solutions of
the given ODE. Calculate the rst four non-zero terms in each case.
(b) Find a polynomial solution when = 0;2;8 and 10.
Problem 5: For the following ODEs, nd a lower bound for the radius of convergence of
the series solution about the point x0.
(a) (x2 3)y00 +y0 + tan(x)y = 0 about x0 = 0.
(b) (4x2 + 9)y00 +x3y0 +y = 0 about x0 = 2..
Problem 6: Find the singular points of the following ODEs. For each singular point
determine whether they are regular or irregular. For each regular singular point, determine
the indicial equation and the exponents at the singularity.
(a) (x2 4)2y00 + 8(x+ 2)y0 +y = 0.
(b) 2(x 1)2(x2 2)3y00 + 2(x2 1)y0 4xy = 0.
(c) 2 tan2(x) y00 sin (x)y0 y = 0.
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2 ASSIGNMENT 1
Problem 7: Consider the ODE:
6x2(1 +x)y00 + 5xy0 y = 0:
(a) Determine the ordinary points, the regular singular points and irregular singular
points in [0;+1)
(b) Determine two values of r such that there are solutions to the given ODE of the
form. y(x) =
+1P
n=0
anxn+r.
(c) Use the series expansion in (b) to determine two independent solutions of the given
ODE. Calculate only the rst three non-zero terms in each case.
Problem 8: Show that x = 0 is a regular singular point of the following ODE.
8x2y00 2xy0 + 3(x+ 1)y = 0:
Calculate the rst three non-zero terms in each of the two independent solutions about
x0 = 0 of the given ODE.