Part II: Second-Order Equations
Existence and Uniqueness
A second order differential equation can be written as
x = f (x,
x, t),
d2 x
dx
where x = 2 and x =
.
dt
dt
Example. Suppose f (t) is an integrable function (e.g. continuous) and consider
d2 x
= f
Selected Differential Equations Concepts
Eulers Method: y n 1 y n hf ( xn , y n ) , n 0, 1, 2,
Integrating Factor for Linear 1st Order:
Exact 1st Order:
Wronskian:
( x) e P ( x ) dx
M ( x, y )dx N ( x, y )dy 0
W [ f1 , f 2 , f n ]
f
f1
f1
f2
f 2
( n 1)
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First Order Differential Equations
In this section, we study a general method to solve linear first order differential equations. Then, we look at a special form of nonlinear equation.
1. Linear equations:
We write the general linear first order different
Oscillations
Example (Mass-spring system: harmonic oscillator). We consider a massspring system with mass m and spring having a restoring force given by
F = kx, where k is the spring constant and x is the displacement of the
mass from the rest point of th
Inhomogeneous Linear 2nd -order equations
We now consider inhomogeneous linear second-order equations
L[x] :=
d2 x
dx
+
p(t)
+ q(t)x(t) = f (t)
dt2
dt
(1)
where f (t) 6= 0. A solution xp (t) to (1) is called a particular solution or also
a particular inte
Scalar Autonomous ODEs: The Phase Line
Definition. An ODE x = f (x, t) with f : Rn R Rn is said to be
autonomous if it does not depend explicitly on the independent variable, t in
this case. We can write x = f (x).
Example. Consider the ODE x = ax with sa
Existence and uniqueness of solutions
Given a differential equation IVP:
dx
= f (x, t),
dt
x(t0 ) = x0 .
Consider the following questions:
1. Do all equations such as (1) have a solution?
(1)
2. If a solution exists, is it unique?
Example. Consider the so
Differential Equations: MATH 2060U
Chapter 0: Review of Prerequisites
1. The Derivative of a Function: The derivative of f (x) with respect to
df
x is the function f 0 (x)
and is defined as,
dx
f (x + h) f (x)
h0
h
f 0 (x) = lim
1
2. Differentiation Tech
Part 1: First Order Ordinary Differential Equations
Consider the following examples that will provide some context to how ODEs
fit in with other types of equations that you have seen, and will see later.
1. Algebraic Equations
(a) Scalar algebraic equatio
1. CH1: INTRODUCTION TO DIFFERENTIAL EQUATIONS
0 Be familiar with the terminology; at most, get short answer questions from
here (1 or 2 marks).
2. CH2: FIRSTORDER DES
0 possibly a short question on Eulers Method
0 Direction Fields
0 Separable rstorder DE
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