1. Solve the following differential equations, using the method of separating the variables: WI+x - 4] [y = x - 3x + 3C] a. xy + (1 +x* - 0 dy - x-1 b. 器 dx - 3x?y [y - ec] c. d. y = 3x(1 +y') (tan-ly = x + C] %3D * -1-4 3v (- In|1 - 4v| - Inje|+ C] e. f. tan xdy - cot ydx (Inlsinx = Inisec yl + C)
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![ull MTNSA-Its Go Tim... ? 21:30
( Math practice
ODE Notes.pdf
T- 68 °C
EXERCISE
70
1. Solve the following differential equations, using the method of separating the variables:
a. xy + (1 +x² )»' = 0
I+x = 4]
[y' = x - 3x + 3C]
by
- 3x²y
ly - ec]
c.
d. y = 3x²(1 +y²)
[tan-ly = x³ + C]
e.
*de
(- In[1 – 4v²| - Inþe| + C]
f. tanxdy = cotydx
g. (1 +2y)dx + (4 –x² )dy = 0
h. ysinxec0 dx +y'dy = 0
i. y = (cos?x)(cos²2y)
j. xy' - (1 - y²)+.
2. Solve the following boundary value problems:
[In|sinx| = In/sec y +C]
[-놀tanh-' (솔)- Inl +2y + C]
[y =
[2 tan 2y – 2x – sin 2x = C]
[sin"ly = Infx| + C]
a. (1+x') - x?y
given that y(1) = 2
[ - 4(1 +x')]
b. cosy+ (1 + e*) siny.
given that y(0) = 4
[(1 + e*) secy = 2/2]
c.
given that r(1) = 2
If - loiui = †]
d. y - T
y+y? = x² - 4]
given that y(2) = 0
It cos 2x = + sin 3y –
3. According to Newton's law of cooling, the rate at which a substance cools in moving air is
proportional to the difference between the temperature of the substance and that of the air. If the
temperature of the air is 30 °C and the substance cools from 100 °C to 70 °C in 15 minutes, find
[52,16 min]
4. In a certain culture of bacteria the rate of increase of the number of bacteria is proportional to
sin 2xdx + cos 3ydy -0
given that y($) = Í
e.
the time when the temperature will be 40 *C.
the number present.
a. If it is found that the number doubles in 4 hours, how many times the original amount may
[8]
b. If, for a slightly different clone of this virus, there are 10ª at the end of 3 hours and 4 x 104
be expected at the end of 12 hours?
at the end of 5 hours, how many were there in the beginning?
5. Bacteria in a certain culture increase at a rate proportional to the number present. If the number
N increases from 1000 to 2000 in 1 hour, how many are present at the end of 1,5 hour?[ 2828]
6. In a culture of yeast, the amount A of active yeast grows at a rate proportional to the amount
present. If the original amount A, doubles in 2 hours, how long does it take for the original
amount to triple?
[3,17 hr]
71](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff36bde20-c51e-457c-beb0-4683c113fb5c%2F6b6af2b1-0b9a-4f87-ae93-1c0a9e1f8c0d%2Fwv2caak_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps
