Download Assignment 3 - University of Toronto

University of Toronto at Scarborough
Department of CMS, Mathematics
MAT B44F
2015/16
Problem Set #3
Due date: in tutorial, week of Nov 16, 2015
Do the following problems from Boyce-Di Prima.
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3.5: 7, 9 (9th ed: 5,7)
3.6: 6, 10, 13, 14, 16
5.2 #7, 10
5.3 #11
5.4 #39
1. Find a particular solution yp of each of the following equations.
(a) y 00 + 16y = e3x
(b) y 00 − y 0 − 6y = 2 sin 3x
(c) y 00 + 2y 0 − 3y = 1 + xex
(d) y 00 + y = sin x + x cos x
2. Use the method of variation of parameters to find a particular solution of the following
differential equations.
(a) y 00 + 9y = 2 sec 3x
(b) y 00 − 2y 0 + y = x−2 ex
(c) x2 y 00 − 3xy 0 + 4y = x4
(d) x2 y 00 + xy 0 + y = ln(x)
3. Use the method of undetermined coefficients to find particular solutions of the following
equations:
(a) y 00 + 9y = 4 cos 3x
(b) y 00 + 4y 0 + 4y = 3e−2x + e−x
4. For x > 0, find the general solution of the equation
2x2 y 00 + xy 0 − y = 3x − 5x2 .
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5. Use series methods to solve the differential equation
y 00 + xy = 0.
6. Solve the following initial value problem using power series. First make a substitution
P
of the form t = x − a, then find a solution n cn tn of the transformed differential
equation:
(2x − x2 )y 00 − 6(x − 1)y 0 − 4y = 0; y(1) = 0, y 0 (1) = 1.
7. Consider the equation y 00 + xy 0 + y = 0.
(a) Find its general solution in terms of two power series y1 , y2 in x, where y1 (0) = 1
and y2 (0) = 0.
(b) Use the ratio test to verify that the series y1 and y2 converge for all x.
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(c) Show that y1 is the series expansion of e−x /2 . Use this fact to find a second linearly
independent solution by the method of reduction of order.
8. Determine whether x = 0 is an ordinary point, a regular singular point, or an irregular
singular point. If it is a regular singular point, find the exponents of the differential
equation at x = 0.
(a) xy 00 + (x − x3 )y 0 + (sin x)y = 0
(b) x2 y 00 + (cos x)y 0 + xy = 0
(c) x(1 + x)y 00 + 2y 0 + 3xy = 0
9. Solve the following differential equation by power series methods (the method of Frobenius):
2x2 y 00 + xy 0 − (1 + 2x2 )y = 0
10. Solve the following differential equation by power series methods (the method of Frobenius): 2xy 00 − y 0 − y = 0
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