i" x? if
2 5 5; ? = my wa 2
G§§z§ff§§$ Q??? gw'ka féé/ W *5/55; Va w/
w
the general solutions of the following differential equations:
a) 5 i ) gmyx-l)
da: 3(4 ~ 3;)
Answer: QA//-w xix-v/ZthY/fc
2 3 2
b) _ (sec2m+3:52tanyw;y3) dx+ (x3
Math 302 Dierential Equations
=Practice Final=
12/11/2009
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Maryland 310
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Math 302 Dierential Equations Midterm 2 11/11/2009
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Lecture 24: 7.6 Complex eigenvalues.
Ex Find the solution to the system
]
[ ]
[
[ ]
x1
1 2
a
x = Ax,
where x =
, A=
,
x(0) =
,
x2
2 1
b
Sol 1 First we want to nd the eigenvalues r and eigenvectors x = 0:
(7.6.1)
Ax = rx
(A rI)x = 0.
The eigenvalues are so
Lecture 26: 7.7 The exponential matrix.
We will now nd a nice way to the express the solution to the system
x = Ax,
(7.7.1)
where A is a 22 matrix, analogous to the formula for the solution of one equation.
If A has two nonparallel eigenvectors x(1) and x
Lecture 25: 7.8 Repeated eigenvalues. Recall rst that if A is a 2 2 matrix
and the characteristic polynomial have two distinct roots r1 = r2 then the corresponding eigenvectors x(1) and x(2) are linearly independent for if c1 x(1) +c2 x(2) = 0
it follows
Lecture 30: 9.2-3. We consider a 2 2 nonlinear system
(9.1)
dx
= f (x),
dt
[
]
x1
x=
,
x2
[
]
f1 (x1 , x2 )
f (x) =
.
f2 (x1 , x2 )
We study the stability of the nonlinear system around critical points f (x0 ) = 0 by
approximating with a linear system as
Lecture 27: 7.9 Nonhomogeneous equations. There are several methods in
the book but we will only go over using diagonalization and the exponential matrix.
Diagonalization Suppose that A is an nn matrix with n linearly independent
eigenvectors A (k) = k (k
Lecture 28: 9 Nonlinear Equations and stability. In this chapter we will
consider autonomous 2 2 nonlinear systems
(9.1)
x = f (x),
[
]
x1
where x =
,
x2
[
]
f1 (x1 , x2 )
f (x) =
.
f2 (x1 , x2 )
Autonomous mean that f does not explicitly depend on t (but
Math 302 Differential Equations
Many phenomena in physics and engineering are described by dierential equations.
Lecture 1: 1.1-2. Two Models, Direction elds, Solution curves.
Model I: A Falling object is subject to Newtons second law:
F = ma
where
a=
dv
Lecture 31: 9.2-3 Trajectories. One can sometimes turn an autonomous system
(9.5)
dx/dt = F (x, y),
dy/dt = G(x, y)
into a rst order equation
dy
dy/dt
G(x, y)
=
=
dx
dx/dt
F (x, y)
(9.6)
This rst order equation can sometimes be solved at least implicitly
Lecture 29: 9.1 The Phase plane for linear systems. In this section my
lecture exactly followed the book and the drawings of the phase portrait are the
most important part so I refer to the book for this section.
1
Lecture 23: 7.5 Linear systems of dierential equations.
Ex 2 Find the solution to the system
[ ]
[
]
[ ]
x1
6 2
a
x = Ax,
where x =
, A=
,
x(0) =
,
x2
2 9
b
Sol First we want to nd the eigenvalues r and eigenvectors x = 0:
(7.5.5)
Ax = rx
(A rI)x = 0.
The
Lecture 21 7.2-3 Algebraic Systems.
Ex. 1 We want to solve the 3 3 system
x1 2x2 + x3 = 0
2x2 8x3 = 8
4x1 + 5x2 + 9x3 = 9
Geometrically this represents the intersection of 3 planes. To minimize the writing
it is convenient to only write out the coecient m
Lecture 22 7.3 Eigenvectors. The equation
(7.3.1)
Ax = y
or
a11 x1 + a12 x2 = y1
a21 x1 + a22 x2 = y2
can be viewed as a map transforming the vector x R2 into the vector y R2 .
Examples of such transformations are scalar multiplication or rotations of a v
Lecture 2: Section 1.2 Analytical solution of the simple models. We can
actually solve the mice-owl model (1.1.3) analytically:
(1.2.1)
dp
= 0.5p 450
dt
The general idea is that we will try to rewrite it as a an equation of the form
(1.2.2)
d
G(t, p(t) =
Lecture 3: 2.1 First order linear equations: Integrating Factor. In this
lecture we will learn to solve a general rst order linear dierential equation
dy
+ p(t)y = g(t).
dt
(2.1.1)
We want to nd a way to write this equation in the form
d
G(t, y(t) = f (t)
Lecture 4: 2.2 Separable equations. We can not solve a general rst order
equation:
dy
(2.2.1)
= f (x, y)
dx
but there is another special case that we can deal with called separable equations.
First we note that there are many ways to write (2.2.1) in the
Lecture 5: 2.3 More models.
Model III: Mixing of chemicals. Suppose a time t = 0 a tank contains Q0 lb
of salt dissolved in 100 gal of water. Assume that water containing 1 lb of salt/gal
4
is entering the tank at a rate of r gal/min, and that the well-st
Lecture 6: 2.4 Dierence between linear and nonlinear dierential equations. For linear equations we have the following existence theorem:
Th 1 Suppose that p and g are continuous functions on an open interval I : <
t < containing t0 . Then there is a uniqu
Lecture 7: 2.5 Autonomous equations and population dynamics. A differential equation is called autonomous if it has the form
(2.5.1)
dy
= f (y),
dt
y(0) = y0
The simplest model of variation of the population of a spices is exponential
growth that the rate
Lecture 8: 3.1: Second order linear dierential equations. We are now
going to study the initial value problem for second order linear dierential equations:
(3.1.1)
y + p (t) y + q(t) y = g(t),
y(t0 ) = y0 ,
y (t0 ) = y0
Such equations are likely to show u
Lecture 9: 3.2 Fundamental Solutions of linear homogeneous equations. Most
of what we will do in this chapter concerns linear second order dierential equations with constant coecients. However, the results in this section also holds for
variable coecients