Chapter 6: Numerical Methods
6.1 Euler Method
Basic Idea
ODE: y = f (t, y )
Assume y (t) is known
For small h approximate
y (t + h) y (t)
y (t)
h
= f (t, y (t)
y (t + h) yap(t + h)
where
yap(t + h) = y (t) + h f (t, y (t)
Truncation Error:
|y (t + h
7.2: Linear Systems with Two or Three Variables
I. Geometry of Solutions
y
y
I.1 One equation for 2 unknowns
Ex.:
2x + y = 2
2
(1)
y=22x
T
p =[0,2]
p+v
x
Solutions form straight line:
y = 2 2x
p+2 v
T
v =[1,2]
(2)
Vector form of (2):
x
y
2t + (2 2t) = 2
=
7.4: Homogeneous and Inhomogeneous Systems
Homogeneous Systems
Form:
Ax = 0
(1)
(1) has always solution x = 0
Any solution x with x = 0 is
called nontrivial
Nontrivial solutions exist if
REF (A) or RREF (A) has a
free column
Let p: pivots
f : free var
Chapter 3: Modelling and Applications
Principle: Develop model function f (t, x) for the rate of change of a
variable x First order ODE: x = f (t, x)
3.3: Personal Finance (Savings Accounts and Loans)
P(t): balance on a savings
account or loan; unit: $;
Chapter 3: Modelling and Applications
Principle: Develop model function f (t, x) for the rate of change
of a variable x First order ODE: x = f (t, x)
3.1: Modelling Population Growth
P (t): Population of species
(bacteria, US-pop., . . . )
Model:
dP/dt
2.3: Models of Motion: Gravity Force and Air Resistance
Gravity Force: Fg = mg
- g = 32 f t/s2 = 9.8 m/s2
Air Resistance: Fair = R(v)
Linear Model:
x
R(v)>0
v>0
R(v ) = kv
- [k] = mass/time
- valid for small velocities
Quadratic Model:
m
m
mg
R(v)<0
R
2.2: Solutions to Separable Equations
Form: dy = g (t)f (y )
dt
Implicit Solution:
[1/f (y )]dy = g (t)dt
[1/f (y )]dy =
g (t)dt ()
Solve () for y explicit solution
Note: () may have several solutions.
Use IC to choose the right one.
Ex.:
dy
dt
= ty 2
or
2.4: Linear Equations
General Form:
x = a(t)x + f (t)
(1)
If f (t) = 0, (1) is called homogeneous:
x = a(t)x
If f (t) = 0, (1) is called nonhomogeneous
Examples of linear equations:
x = sin(t)x
homogenous, a(t) = sin t
x = x/t
homogenous, a(t) = 1/t
y = e
2.7: Existence and Uniqueness of Solutions
Basic Existence and Uniqueness Theorem (EUT):
Suppose f (t, x) is dened and continuous, and has a continuous
partial derivative f (t, x)/x on a rectangle R in the txplane.
Then, given any initial point (t0, x0) i
2.9: Autonomous Equations and Stability
Form:
x = f (x)
Ex: x = sin(x), y = y 2 + 1
Implicit Solution:
[1/f (x)] dx =
are autonomous
dt
x = sin(tx), y = xy
are not autonomous
G(x) = t + C
where G(x) = [1/f (x)] dx is an
antiderivative of 1/f (x)
Conseque
Fall 11, MATH 345: Sample Final Problems
1: Classify the following dierential equations as nonlinear and separable, nonlinear and
non-separable, linear homogeneous, or linear inhomogeneous. Note: You dont need to
solve the equations, just classify!
(a)
t
M 345 Assignment 2
solutions 1. Use variation of parameters to solve for y(t)
1 + t4 y 0 (t) + 12t3 y (t) = 1 + t4
y (1) = 2
2
;
Rewrite the equation in normal form
y 0 (t) +
12t3
y (t) = 1 + t4
1 + t4
3
;
(a) nd the general homogeneous solution and expla
Exam 1
M 317 sec 2
1. Explain why division by zero is an illegal operation.
2. Explain why a number whose decimal representation is repeating or eventually repeating
must be a rational number.
3. Explain why a rational number will always have a decimal re
M 345
1.
dy
dt
Problem set 1
Solutions
yt
a)
direction field (autonomous)
The term "autonomous" implies the direction arrows vary with y but do not depend on t.
b)
dy dt
2 y tC
so
y
yt
and
1
2
tC
2
For each choice of the constant C the function y t is a
M345 Exam I Find the general solution to each equation. If initial conditions
are given, nd the unique solution to the initial value problem.
1. (x
y ) dx
(x + y ) dy = 0
Since @y (x y ) = 1
the equation is exact
If
@x F = x
If
@y F =
Evidently
2.
and
y ;