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

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