ODE HOMEWORK 2
DUE THURSDAY, JUNE 5
Monday, June 3
• §3.1 #6, 7
• §3.2 #17, 21
Problem 1. Consider the equation
y 00 + p(t)y 0 + q(t)y = 0,
(1)
where p(t), q(t) are continuous for t ∈ (−1, 1). Define the set
Y := φ : (−1, 1) → Rφ is a solution of (1) .
(i) Show that Y is a vector space of dimension at least 1. That is, show that the
set Y contains at least one non-zero element element, and if φ1 , φ2 ∈ Y then
aφ1 + bφ2 ∈ Y for any constants a, b ∈ R.
(ii) Let φ1 , φ2 ∈ Y, and let W (t) be the Wronskian determinant of φ1 (t), φ2 (t). Show
that W satisfies the equation
W 0 + p(t)W = 0.
Conclude that if there exists a t0 ∈ (−1, 1) such that W (t0 ) 6= 0 then W (0) 6= 0.
(iii) Suppose that the Wronskian determinant W (t) satisfies W (t0 ) 6= 0 for some t0 ∈
(−1, 1). Prove that the set {φ1 , φ2 } is a basis for the vector space Y, and in particular, Y has dimension 2 as a real vector space.
• §3.3 # 14, 18
• §3.4 # 13, 14, 28
Problem 2. The aim of this problem is to prove Theorems 3.5.1 and 3.5.2 in the book.
Consider the linear second order ordinary differential equation
(2)
y 00 + p(t)y 0 + q(t)y = g(t).
(i) Assume that Y1 (t) and Y2 (t) are solutions to equation (2). Show that Y1 (t) − Y2 (t)
is a solution of the associated homogeneous equation
(3)
y 00 + p(t)y 0 + q(t)y = 0.
(ii) Assume that y1 (t), y2 (t) form a fundamental set of solutions to equation (3). Argue
that
Y1 (t) − Y2 (t) = c1 y1 (t) + c2 y2 (t)
1
2
DUE THURSDAY, JUNE 5
for some constants c1 , c2 . (Hint: we did this in class)
(iii) Show that the function
φ(t) = c1 y1 (t) + c2 y2 (t) + Y2 (t),
is a general solution of equation (2). That is, show that for any initial values
y(0) = y0 ,
y 0 (0) = y00
we can find constants c1 , c2 so that φ(t) solves (2) with initial values φ(0) = y0 and
φ0 (0) = y00 .
Tuesday, June 4
• §3.5 # 8, 15, 18. In these problems, Table 3.5.1 on page 181 may be of use. You
should try to understand why the “guesses” in said table make sense. The table
will not be available to you on the exams.
Problem 3. The aim of this problem is to discuss (in an example) the failure of the method
of undetermined coefficients. Consider the equation
(4)
y 00 − 3y 0 + 2y = e2t .
Try to find a general solution of this equation using the method of undetermined coefficients.
Why does this method fail?
Wednesday, June 5
• §3.6 #7, 8, 9, 17
Problem 4. Find a solution to equation (4) using the method of variation of parameters.
BONUS REVIEW PROBLEMS
The problems listed below are review problems for Chapter 2. These are not mandatory,
but some bonus points will be given for completing all of them. Moreover, it’s good practice!
• §2.1 # 15, 19
• §2.2 # 24
• §2.4 # 3, 15
• §2.5 # 9
• §2.6 # 15