Differential Equations, Probability, and Statistics
M 2A

Spring 2014
90
PART IV: QUALITATIVE ANALYSIS OF 2D AUTONOMOUS SYSTEMS
32. Orbits of autonomous systems
32.1. Autonomous systems.
x = v (x),
x(t) D Rn .
The domain D (which can be the whole space Rn ) is the phase space of the system.
We interpret the given vectorval
Differential Equations, Probability, and Statistics
M 2A

Spring 2014
36
PART II: LINEAR EQUATIONS
15. Basic concepts
15.1. Linear equations. The standard form of a second order linear equation is
L[x] x + p(t)x + q (t)x = g (t).
The map x L[x] is a dierential operation. If g (t) 0, then the equation
L[x] = 0
is called homo
Differential Equations, Probability, and Statistics
M 2A

Spring 2014
56
PART III: LINEAR EQUATIONS (ADDITIONAL TOPICS)
23. Linear equations with analytic coefficients
Well be considering equations
y + p(x)y + q (x)y = 0
(23.1)
such that both coecients p and q are analytic in some neighborhood of x0 .
23.1. Power series sol
Differential Equations, Probability, and Statistics
M 2A

Spring 2014
Ma2a (analytical)
Fall 2012
PART I: FIRST ORDER ODEs
1. Introduction
1.1. Fundamental theorem of calculus. The simplest (trivial) d.e.:
y (x) = f (x).
Examples:
(i) Find all (maximal) solutions of the equation
1
.
y (x) =
1 + x2
Answer:
dx
y (x)
= arctan
Differential Equations, Probability, and Statistics
M 2A

Spring 2014
MATH 2A 2013 HOMEWORK 8 SOLUTIONS
1
Problem 110 pts (Problem 34.1(ii) For potential function V (x) = 2 x4 x2 , write down the total energy for a particle
of unit mass, and assuming that this is conserved write down a coupled system for x and y = x.
Draw
Differential Equations, Probability, and Statistics
M 2A

Spring 2014
MATH 2A 2013 HOMEWORK 7 SOLUTIONS
Problem 110 pts
1
(Problem 28.6) Suppose that A has two eigenvalues 1 = 0 with eigenvector v1 and 2 = 0 with
eigenvector v2 .
(i) Write down the general solution to the equation x = Ax.
(ii) After changing to a coordinat