Economics 6215
Columbia University
Professor Xavier
Problem Set 1.
Problem 1.
Find the General Solution and the Particular Solution for each
of the following Linear Autonomous Differential Equations.
(A) 
with y(0)=2
(B) 
with y(0)=3
Problem 2.
Find the General Solution and the Particular Solution for each
of the following Linear Non-Autonomous Differential Equations.
(A) 
with y(0)=3/2
(B) 
with y(0)=e.
Problem 3.
For each of the following differential equations, display a
phase diagram, find the steady state(s) and describe the
qualitative dynamics of the variable for any initial condition.
(A)
(Compare with your answer in 1.A)
(B) 
, where A>0, 
,
and 
are constant numbers.
(C) 
where A>0, 
,
and 
. Consider the case when 
and the case when 
.
Problem 4.
For each of the following system of ODEs display a phase
diagram for the system of ODEs and discuss stability.
(A) 
(B) 
Problem 5.
Consider the following non-linear systems of ODEs. For each of
them draw a phase diagram and establish the stability properties
of the steady-state(s)
(A) 
(B) 
Problem 6.
Consider the following dynamic model (do not try to interpret
this economically, yet):
Max 
subject to 
and 
.
Find the set of all first order conditions.