Practical class № 11. Differential equations of the first and second order. Homogeneous and non-homogeneous linear differential equations
Theoretical questions: 1. Definition of differential equations; 2. Equations with separable variables; 3. First order linear differential equation; 4. Higher order differential equations; 5. Homogeneous second-order equations; 6. Non-homogeneous second-order equations
Classroom assignments:
1. Solve differential equations:
1.
;
2.
3.
4.
5.
6.
7.
.
8.
9.
10.
2. Find the particular solutions of differential equations:
а)
,
б)
,
2. Find the general solution of the differential equation:
1. y-2y+5y=0 |
2. y-y=0 |
3. y-4y+3y=0 |
4. y-4y+4y=0 |
5.
|
6.
|
7.
|
8.
|
9.
|
10.
|
11
|
12.
|
13.
|
14.
|
15.
|
16.
|
17.
|
18.
|
3. Find the solution of non-homogeneous second order differential equations:
1. |
6.
|
11.
|
2.
|
7.
|
12.
|
3.
|
8.
|
13.
|
4.
|
9.
|
14.
|
5.
|
10.
|
|
Homework:
Theoretical material: Numerical series. The convergence of the sum of the series.
Solve problems:
1. Solve differential equations:
1.
2.
3.
4.
5.
6.
7.
8.
2. Find the particular solutions of equations:
,
.
3.
Solve equations: 1.
,
,
.
2.
,
,
.
3.
,
,
.
4.
,
,
.
5.
,
,
.
6.
,
,
.
7. , , .
8.
,
,
.
9.
,
,
.
PRACTICAL CLASS № 12.
Numerical series. The convergence of the sum of the series. Alternating series. Absolute and conditional convergence of the series. Functional series. The area of series convergence. Power series. The radius of convergence
Theoretical questions:
1. Numerical series
2. The convergence of the sum of the series
3. Alternating series
4. Functional series
5. Power series
Classroom assignments:
Find at least 10 partial sums of the given series. Is it convergent or divergent? Expalin.
a).
;
b).
.
2. Determine whether the series is convergent or divergent. Find sum of series if it’s convergent.
a).
;
b).
.
3. Test the convergence of the series
a)
;
b)
;
c)
;
4. Find the radius of convergence and interval of convergence of the power series
a)
;
b).
;
c).
.
Homework:
Theoretical material: Basic concepts of probability theory. Addition and multiplication theorems of probabilities.
Solve problems:
1. Determine using integral test whether the series convergent or divergent
a).
;
b).
;
c).
.
2. Test the following series:
a).
;
b).
.
c)
is
convergent d)
is
convergent e)
is
divergent
f)
is
divergent g)
is
divergent h)
is
convergent (Hint: consider
a convergent geometric series with
,
)
i)
is
convergent j)
is convergent
3. Test the convergence of the series
a)
;
b)
;
c)
.
4. Test the convergence of the series
a)
;
b)
.
5. Apply ratio test to verify that the given series are absolutely convergent and thereby convergent
a)
b)
c)
d)
e)
Find a power series representation of the function and determine interval of convergence.
a).
;
b).
;
c).
.
PRACTICAL CLASS № 13-14.
Basic concepts of probability theory. The classification of events. The classical definition of the event probability. Addition and multiplication theorems of probabilities. The formula of total probability. The formula of Bernoulli, Poisson's ratio
Theoretical questions:
Main concept. Definition of probability
Properties of probability
Classroom assignments:
1. In a workshop there are 4 kinds of beds, 3 kinds of closets, 2 kinds of shelves and 7 kinds of chairs. In how many ways can a person decorate his room if he wants to buy in the workshop one shelf, one bed and one of the following: a chair or a closet?
2.
If we throw a fair die, any one of the six
numbers 1, 2, 3, 4, 5 and 6 may come up. There are altogether 6
possible outcomes and the occurrence of each of these outcomes is
equally likely. Hence, we conclude that each number has an equal
chance to appear and the probability of throwing, say, the number ‘1’
is equal to
.
The
probability for the event: ‘number 5 comes up’ is denoted by
.
In fact,
.
Think!
In a single throw of a die, what is the probability of getting
an odd number?
an even number?
a prime number?
a number greater than 4?
a number from 1 to 6?
a number except 5?
3. Three people are to be seated on a bench. How many different sitting arrangements
are possible if Erik must sit next to Joe?
4. How many 3-digit numbers satisfy the following conditions: The first digit is
different from zero and the other digits are all different from each other?
5. Barbara has 8 shirts and 9 pants. How many clothing combinations does Barbara
have, if she doesn’t wear 2 specific shirts with 3 specific pants?
6. A credit card number has 5 digits (between 1 to 9). The first two digits are 12 in
that order, the third digit is bigger than 6, the forth is divisible by 3 and the fifth digit
is 3 times the sixth (which is any number bet 1-9). How many different credit card
numbers exist?
7. In jar A there are 3 white balls and 2 green ones, in jar B there is one white ball
and three green ones. A jar is randomly picked, what is the probability of picking up a
white ball out of jar A?
Homework:
Theoretical material: Random variables, their types. Numerical characteristics of continuous random variables.
Solve problems:
1. If a card is drawn from a pack of 52 playing cards, any one of 52 cards may come up. There are altogether 52 possible outcomes and the occurrence of each of these outcomes is equally likely.
Think!
If we draw a card from a pack of 52 playing cards, what is the probability of getting
the ace of diamonds?
a card of heart?
a card of number 8?
2. When a baby is born, the probability that it is a boy or a girl is theorectially equal,
i.e.
.
Think!
If a woman has three daughters, what is the probability that the next baby is a boy?
3. When a card is drawn from a pack of 52 playing card, what is the probability of getting
(a) an ace and a heart?
(b) an ace or a heart?
4. Out of a box that contains 4 black and 6 white mice, three are randomly chosen.
What is the probability that all three will be black?
5. There are 18 balls in a jar. You take out 3 blue balls without putting them back
inside, and now the probability of pulling out a blue ball is 1/5. How many blue balls
were there in the beginning?