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.
S. 3.5: 7, 9 (9th ed: 5,7)
S. 3.6: 6, 10, 13, 14, 16
S. 5.2 #7, 10
S. 5.3 #11
S. 5.4 #39
1. Find a particular solution yp of each of the following equations.
(a) y
00 + 16y = e
3x
(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
e
x
(c) x
2
y
00 – 3xy0 + 4y = x
4
(d) x
2
y
00 + xy0 + 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
2x
2
y
00 + xy0 – y = 3x – 5x
2
.
1
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
of the form t = x – a, then find a solution P
n cnt
n of the transformed differential
equation:
(2x – x
2
)y
00 – 6(x – 1)y
0 – 4y = 0; y(1) = 0, y0
(1) = 1.
7. Consider the equation y
00 + xy0 + 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.
(c) Show that y1 is the series expansion of e
-x
2/2
. 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) xy00 + (x – x
3
)y
0 + (sin x)y = 0
(b) x
2
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):
2x
2
y
00 + xy0 – (1 + 2x
2
)y = 0
10. Solve the following differential equation by power series methods (the method of Frobenius):
2xy00 – y
0 – y = 0
2

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.
S. 3.5: 7, 9 (9th ed: 5,7)
S. 3.6: 6, 10, 13, 14, 16
S. 5.2 #7, 10
S. 5.3 #11
S. 5.4 #39
1. Find a particular solution yp of each of the following equations.
(a) y
00 + 16y = e
3x
(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
e
x
(c) x
2
y
00 – 3xy0 + 4y = x
4
(d) x
2
y
00 + xy0 + 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
2x
2
y
00 + xy0 – y = 3x – 5x
2
.
1
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
of the form t = x – a, then find a solution P
n cnt
n of the transformed differential
equation:
(2x – x
2
)y
00 – 6(x – 1)y
0 – 4y = 0; y(1) = 0, y0
(1) = 1.
7. Consider the equation y
00 + xy0 + 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.
(c) Show that y1 is the series expansion of e
-x
2/2
. 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) xy00 + (x – x
3
)y
0 + (sin x)y = 0
(b) x
2
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):
2x
2
y
00 + xy0 – (1 + 2x
2
)y = 0
10. Solve the following differential equation by power series methods (the method of Frobenius):
2xy00 – y
0 – y = 0
2

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.
S. 3.5: 7, 9 (9th ed: 5,7)
S. 3.6: 6, 10, 13, 14, 16
S. 5.2 #7, 10
S. 5.3 #11
S. 5.4 #39
1. Find a particular solution yp of each of the following equations.
(a) y
00 + 16y = e
3x
(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
e
x
(c) x
2
y
00 – 3xy0 + 4y = x
4
(d) x
2
y
00 + xy0 + 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
2x
2
y
00 + xy0 – y = 3x – 5x
2
.
1
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
of the form t = x – a, then find a solution P
n cnt
n of the transformed differential
equation:
(2x – x
2
)y
00 – 6(x – 1)y
0 – 4y = 0; y(1) = 0, y0
(1) = 1.
7. Consider the equation y
00 + xy0 + 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.
(c) Show that y1 is the series expansion of e
-x
2/2
. 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) xy00 + (x – x
3
)y
0 + (sin x)y = 0
(b) x
2
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):
2x
2
y
00 + xy0 – (1 + 2x
2
)y = 0
10. Solve the following differential equation by power series methods (the method of Frobenius):
2xy00 – y
0 – y = 0
2