MATH 557 Homework Set #1
Fall 2015
1. Give a correct explanation of why the separation of variables method works for solving
dx
P (t)
=
dt
Q(x)
2. If we use the separation of variables method to solve
dx
= x2
dt
do we obtain all possible solutions? Discuss.
3. Find the general solution of the third order Euler type equation
(t − 1)3 φ000 + 8(t − 1)2 φ00 + 14(t − 1)φ0 + 6φ = 0
4. If a, b, c > 0 show that any solution of aφ00 + bφ0 + cφ = 0 satisfies limt→+∞ φ(t) = 0.
5. An ODE is said to be of Bernoulli type if it can be written as
φ0 + a(t)φ = b(t)φn
for some real exponent n. Show that the Bernoulli equation can be transformed to a
first order linear equation by means of the change of variable ψ = φ1−n . Find the general
solution of
tφ0 + φ = t4 φ3
1 1
. (It is not too
6. Find the general solution of the system φ = Aφ where A =
0 1
hard to compute etA directly from its power series definition in this case.)
0