The University of British Columbia
MATH 253
Midterm 1
9 October 2013
Time: 50 minutes
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LECTURE 33: APPLICATIONS OF NONLINEAR SYSTEMS
MINGFENG ZHAO
November 24, 2014
Conservative equations
For a conservative equation x + f (x) = 0, let y = x , then we have the system:
x = y,
and y = f (x).
Then there are never any asymptotically stable point
LECTURE 34: APPLICATIONS OF NONLINEAR SYSTEMS
MINGFENG ZHAO
November 24, 2014
Predator-prey or Lotka-Volterra systems
Let
x =
number of the prey(e.g., hares)
y
number of the predator(e.g., foxes)
=
Recall the population model: the rate of growth of the po
LECTURE 32: STABILITY AND CLASSIFICATION OF ISOLATED CRITICAL POINTS
MINGFENG ZHAO
November 21, 2014
Stability and classication of isolated critical points
The following table shows the behavior of an almost linear system near an isolated critical point:
LECTURE 30: LINEARIZATION, CRITICAL POINTS, AND EQUILIBRIA
MINGFENG ZHAO
November 17, 2014
Autonomous systems
x = f1 (x1 , x2 )
1
The vector eld/phase portrait/phase diagram of
is the projection on the x1 x2 -plane of its three
x = f (x , x )
2 1
2
2
x
LECTURE 27: TWO DIMENSIONAL SYSTEMS AND THEIR VECTOR FIELDS
MINGFENG ZHAO
November 10, 2014
In this course, we only study the two dimensional homogeneous system:
x = f1 (x1 , x2 )
1
x = f (x , x )
2 1
2
2
x = f1 (x1 , x2 )
1
The vector eld of
is the pr
LECTURE 25: EIGENVALUE METHOD AND MULTIPLE EIGENVALUE
MINGFENG ZHAO
November 03, 2014
Theorem 1. Let A be a 2 2 matrix, y(t) and z(t) be two linearly independent solutions to x = Ax, then the general
solution to x = Ax is
x(t) = C1 y(t) + C2 z(t).
In this
LECTURE 31: STABILITY AND CLASSIFICATION OF ISOLATED CRITICAL POINTS
MINGFENG ZHAO
November 19, 2014
Linearization
x = f (x, y)
Denition 1. Suppose (x0 , y0 ) is an equilibrium solution to the autonomous system
. Let u = x x0
y = g(x, y)
x = f (x, y)
a
LECTURE 28: NONHOMOGENEOUS SYSTEMS
MINGFENG ZHAO
November 12, 2014
For the the two dimensional autonomous linear system x = Ax, where A is a 2 2 constant matrix. Assume A has
two distinct nonzero eigenvalues 1 = 2 , the behavior of solutions to the two di
LECTURE 29: NONHOMOGENEOUS SYSTEMS
MINGFENG ZHAO
November 14, 2014
Undetermined coecients
Let A be a 2 2 matrix, consider the system:
x = Ax + f (t).
Let a, b be two constant vectors, pn (t) and pn (t) be polynomials with degree n. If f (t) has the form:
LECTURE 26: MULTIPLE EIGENVALUE, TWO DIMENSIONAL SYSTEMS AND THEIR
VECTOR FIELDS
MINGFENG ZHAO
November 07, 2014
Theorem 1. Let A be a 22 matrix with two distinct eigenvalues 1 and 2 , and v1 and v2 are eigenvectors corresponding
to 1 and 2 , respectively
LECTURE 23: SYSTEMS OF ODES
MINGFENG ZHAO
October 29, 2014
Introduction to systems of ODEs
In this course, we only study the systems of the rst order dierential equations which has the form:
x (t) = f1 (t, x1 , x2 )
1
x (x) = f (t, x , x ).
2
2
1
2
A se
LECTURE 24: EIGENVALUE METHOD
MINGFENG ZHAO
October 31, 2014
Theorem 1. Let A be a 2 2 matrix, y(t) and z(t) be two linearly independent solutions to x = Ax, then the general
solution to x = Ax is
x(t) = C1 y(t) + C2 z(t).
Theorem 2 (Eigenvalue Method). L
LECTURE 20: TRANSFORMS OF DERIVATIVES AND ODES
MINGFENG ZHAO
October 22, 2014
Proposition 1. There holds that
I. Linearity:
L[af (t) + bg(t)](s) = aL[f (t)](s) + bL[g(t)](s).
That is,
L1 [aF (s) + bG(s)](t) = aL1 [F (s)](t) + bL1 [G(s)](t).
II. First Shif
LECTURE 22: DIRAC DELTA AND IMPULSE RESPONSE
MINGFENG ZHAO
October 27, 2014
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est dt,
L[f ](s) =
for all s > 0.
0
II. If F (s) = L[f ](
LECTURE 21: CONVOLUTION
MINGFENG ZHAO
October 24, 2014
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est dt,
L[f ](s) =
for all s > 0.
0
II. If F (s) = L[f ](s), the inverse Lapla
LECTURE 18: THE LAPLACE TRANSFORM
MINGFENG ZHAO
October 17, 2014
Example 1. A mass of 4 kg on a spring with k = 4 and a damping constant c = 1. Suppose F0 = 2. Using forcing
function F0 cos(t), nd the that causes practical resonance and nd the amplitude.
LECTURE 19: TRANSFORMS OF DERIVATIVES AND ODES
MINGFENG ZHAO
October 20, 2014
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est dt,
L[f ](s) =
for all s > 0.
0
II. If F (s) = L[f
LECTURE 15: NONHOMOGENEOUS EQUATIONS
MINGFENG ZHAO
October 08, 2014
Undetermined Coecients:
Let a, b and c be constants, consider the equation:
ay + by + cy = f (x).
Let pn (x) and pn (x) be polynomials with degree n, a particular solution yp (x) to ay +
LECTURE 17: FORCED OSCILLATIONS AND RESONANCE
MINGFENG ZHAO
October 15, 2014
Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx + cx + kx = F (t) .
We are interested in periodic forcing, that is, F (t) = F0
LECTURE 16: FORCED OSCILLATIONS AND RESONANCE
MINGFENG ZHAO
October 10, 2014
Undetermined Coecients:
Let a, b and c be constants, consider the equation:
ay + by + cy = f (x).
Let pn (x) and pn (x) be polynomials with degree n, a particular solution yp (x)
LECTURE 14: NONHOMOGENEOUS EQUATIONS
MINGFENG ZHAO
October 06, 2014
Theorem 1. Let y1 (x) and y2 (x) be a fundamental set of solutions to the homogeneous equation y +p(x)y +q(x)y = 0,
and yp (x) be any particular solution to the nonhomogeneous equation y
LECTURE 13: NONHOMOGENEOUS EQUATIONS
MINGFENG ZHAO
October 03, 2014
Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx + cx + kx = F (t) .
For free motion (that is, F (t) = 0), rewrite the equation, we have
LECTURE 12: MECHANICAL VIBRATIONS
MINGFENG ZHAO
September 29, 2014
Recall in the last lecture:
I. Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx + cx + kx = F (t) .
II. RLC Circuit System:
Figure 2. RLC
LECTURE 6: EXACT EQUATIONS AND INTEGRATE FACTORS
MINGFENG ZHAO
September 15, 2014
Recall that
I. y = f (x)g(y) =
1) y(x) a for some constant a such that g(a) = 0
2)
1
dy =
g(y)
II. y + p(x)y = f (x) = r(x) = e
p(x) dx
y + p(x)y = f (x)
III.
= r(x) = e
LECTURE 9: SECOND ORDER LINEAR ODES
MINGFENG ZHAO
September 22, 2014
To draw the phase diagram of y = f (y):
1) Find all critical points of f (y), that is, nd all zeros of f (y).
2) Mark all critical points on a vertical line.
3) For any two neighboring c
LECTURE 11: MECHANICAL VIBRATIONS
MINGFENG ZHAO
September 26, 2014
To nd the general solution to a constant coecient second order liner dierential equation ay + by + cy = 0:
1) Write the characteristic equation of ay + by + cy = 0:
ar2 + br + c = 0.
2) Fi
LECTURE 10: CONSTANT COEFFICIENT SECOND ORDER LINEAR ODES
MINGFENG ZHAO
September 24, 2014
Theorem 1. Let p(x) and q(x) be continuous functions, y1 and y2 are two linearly independent solutions to a homogeneous equation y + p(x)y + q(x)y = 0. Then the gen
LECTURE 5: LINEAR EQUATIONS AND THE INTEGRATING FACTOR
MINGFENG ZHAO
September 12, 2014
2
2
Example 1. Solve y = x2 ex y 2 ex + 1, y(1) = 1.
Rewrite the equations:
2
y = (x2 y 2 )ex + 1,
2
y(1) = 1.
2
2
2
Its easy to see that y(x) = x is a solution to y =