Math 324 Homework 2 Solutions
Instructor: J. Metcalfe
Section 7.1, # 39. The unit staircase function is dened as follows:
f (t) = n if n 1 t < n, n = 1, 2, 3 . . .
a. Sketch the graph of f to see why its name is appropriate.
b. Show that
f (t) =
u(t n)
n=
Math 302 - Dierential Equations (Metcalfe)
Summer 2001 June 26, 2001
Series Solutions Near an Ordinary Point (Section 5.2) When: This technique can be used to solve linear homogeneous dierential equations near an ordinary point. It can, also, be used in c
Math 302 - Dierential Equations (Metcalfe)
Summer 2001 June 5, 2001
Reduction of Order When: We know that the general solution of a second-order, linear homogeneous dierential equation consists of two independent pieces. If we know one of these two indepe
Math 302 - Dierential Equations (Metcalfe)
Summer 2001 June 18, 2001
Method of Undetermined Coecients (Section 3.6, 4.3) When: Use this technique to solve linear nonhomogeneous equations when the forcing term consists of combinations of polynomials, sines
Math 302 - Dierential Equations (Metcalfe)
Summer 2001 June 3, 2001
Solving Linear, Homogeneous Second-Order Equations with Constant Coecients (Sections 3.1,3.4 When: Use this technique for linear homogeneous second-order equations with constant coecients
Math 302 - Dierential Equations (Metcalfe)
Summer 2001 May 31, 2001
Solving Separable Equations (Section 2.2) When: Use this technique for rst-order separable equations. A separable equations is one that can be written in the form: N (y) dy = M (x) dx The
Math 324 Homework 10 Solutions
Instructor: J. Metcalfe Section 8.3 # 20 Find two linearly independent Frobenius series solutions (for x > 0) of 3xy + 2y + 2y = 0. We first notice that x0 = 0 is a regular singular point. So, let y = We compute y and y from
Math 324 Homework 6 Solutions
Instructor: J. Metcalfe
3 Section 7.4, # 26 Find the inverse transform of F (s) = tan1 s+2 .
We apply Theorem 2: 1 f (t) = L1 cfw_F (s) t 1 3 1 = L1 3 2 (s + 2)2 t 1 + ( s+2 ) 3 1 = L1 t (s + 2)2 + 32 1 3 = e2t L1 2 + 32 t s
Math 524 - Elementary Dierential Equations
Instructor: J. Metcalfe Due: March 27, 2008
Assignment 17 Section 5.4, # 8 Find the general solution to 25 12 0 x = 18 5 0 x. 6 6 13 Since 0 = det A I = ( 13)2 ( 7), we have that = 7 and = 13 are the eigenvalues.
Math 324 Homework 3 Solutions
Instructor: J. Metcalfe Section 7.2, # 34 If f (t) = (1)[t] is the square-wave function (whose graph is given on p. 463 of your book), then 1 s Lcfw_f (t) = tanh s 2 . Derive the Laplace transform. We use the generalization o
Math 302 - Dierential Equations (Metcalfe)
Summer 2002
June 23, 2001
Final Exam - Practice Exercises
1. Write
3y (4) 5y + 9y = 6et 2t
as a system of linear, rst order dierential equations.
2. Find the general solution to the system of equations:
dx
= 4x +
Math 324 Homework 3 Solutions
Instructor: J. Metcalfe
Section 7.2, # 34 If f (t) = (1)[t] is the square-wave function (whose graph is given on p. 463
of your book), then
1
s
Lcfw_f (t) = tanh
s
2
.
Derive the Laplace transform.
We use the generalization o
Math 324 Homework 6 Solutions
Instructor: J. Metcalfe
3
Section 7.4, # 26 Find the inverse transform of F (s) = tan1 s+2 .
We apply Theorem 2:
1
f (t) = L1 cfw_F (s)
t
1
3
1
= L1
3 2 (s + 2)2
t
1 + ( s+2 )
3
1
= L1
t
(s + 2)2 + 32
1
3
= e2t L1
2 + 32
t
s
Math 324 Homework 6 Solutions
Instructor: J. Metcalfe
Section 8.1, # 14 Use the method of Example 4 to nd two linearly independent power series
solutions of the dierential equation y + y = x. Determine the radius of convergence of each series,
and identif
Math 324 Homework 12 Solutions
Instructor: J. Metcalfe
8.4 # 10 Find the rst four nonzero terms in a Frobenius series solution of x2 y +xy +(x2 +1)y = 0.
Then use the reduction of order technique (as in Example 4) to nd the logarithmic term and the
rst th
Math 524 - Elementary Dierential Equations (Metcalfe)
Fall 2008
September 25, 2008
Exam 1
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed book exam.
Math 524 - Elementary Dierential Equations (Metcalfe)
Fall 2008
November 6, 2008
Exam 2
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed book exam. N
Math 302 - Dierential Equations (Metcalfe)
Summer 2002
June 4, 2002
Exam 1 - Practice Exercises
1. Find a general solution to
dy
2
+ xy = ex /2
dx
2. Find a solution of:
(3x4 y 1) dx + x5 dy = 0;
y (1) = 1
3. Find a general (implicit) solution of
2
xy 3 d
Math 302 - Dierential Equations (Metcalfe)
Summer 2002
June 17, 2002
Exam 2 - Practice Exercises
1. Find a solution to the initial value problem
y 2xy + 8y = 0;
y (0) = 3,
y (0) = 0
2. Find a general solution to
(x2 1)y + 4xy + 2y = 0
and give a lower bou
Math 324 Homework 6 Solutions
Instructor: J. Metcalfe Section 8.1, # 14 Use the method of Example 4 to nd two linearly independent power series solutions of the dierential equation y + y = x. Determine the radius of convergence of each series, and identif