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3 years ago

(6x-5y+4)dy+(y-2x-1)dx=0

Mathematics

1 answer:

Len [333]3 years ago

8 0

(6x - 5y + 4) dy + (y - 2x - 1) dx = 0

(6x - 5y + 4) dy = (2x - y + 1) dx

dy/dx = (2x - y + 1) / (6x - 5y + 4)

Let X = x - a and Y = y - b. We want to find constants a and b such that

dY/dX = (a rational function)

where the numerator and denominator on the right side are free of constant terms. Substituting x and y in the equation, we have

dY/dX = (2 (X + a) - (Y + b) + 1) / (6 (X + a) - 5 (Y + b) + 4)

dY/dX = (2X - Y + 2a - b + 1) / (6X - 5Y + 6a - 5b + 4)

Then we solve for a and b in the system,

2a - b + 1 = 0

6a - 5b + 4 = 0

==> a = -1/4 and b = 1/2

With these constants, the equation reduces to

dY/dX = (2X - Y) / (6X - 5Y)

Now substitute Y = VX and dY/dX = X dV/dX + V :

X dV/dX + V = (2X - VX) / (6X - 5VX)

The equation becomes separable after some simplification:

X dV/dX + V = (2 - V) / (6 - 5V)

X dV/dX = (2 - V) / (6 - 5V) - V

X dV/dX = (2 - V - (6 - 5V)) / (6 - 5V)

X dV/dX = (4V - 4) / (6 - 5V)

- (5V - 6) / (4V - 4) dV = 1/X dX

Integrate both sides:

-5/4 V + 1/4 ln|4V - 4| = ln|X| + C

Extract a constant from the logarithm on the left:

-5/4 V + 1/4 (ln(4) + ln|V - 1|) = ln|X| + C

-5/4 V + 1/4 ln|V - 1| = ln|X| + C

-5V + ln|V - 1| = 4 ln|X| + C

Get this back in terms of Y :

-5Y/X + ln|Y/X - 1| = 4 ln|X| + C

Now get the solution in terms of y and x :

-5 (y - 1/2)/(x + 1/4) + ln|(y - 1/2)/(x + 1/4) - 1| = 4 ln|x + 1/4| + C

With some manipulation of constants and logarithms, and a bit of algebra, we can rewrite this solution as

-5 (4y - 2)/(4x + 1) + ln|(4y - 4x - 3)/(4x + 1)| = 4 ln|x + 1/4| + 4 ln(4) + C

-5 (4y - 2)/(4x + 1) + ln|(4y - 4x - 3)/(4x + 1)| = 4 ln|4x + 1| + C

-5 (4y - 2)/(4x + 1) + ln|4y - 4x - 3| - ln|4x + 1| = 4 ln|4x + 1| + C

-5 (4y - 2)/(4x + 1) + ln|4y - 4x - 3| = 5 ln|4x + 1| + C

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worty [1.4K]

The line of the best fit is y = 2100x -4166040 and the number of bottles is, 86460 which expect to sell in 2025

A mathematical notion called the line of the best fit connects points spread throughout a graph. It's a type of linear regression that uses scatter data to figure out the best way to define the dots' relationship.

We have data shown in the table:

To find the line of best fit, we will calculate its slope and y-intercept.

We know the slope is given by:

$\rm m = \dfrac{n\sum xy-\sum x \sum y}{n \sum x^2 - (\sum x)^2}$ where n = 5

After calculating, we get,

$\rm n\sum xy-\sum x \sum y = 105000$ and

$\rm n \sum x^2 - (\sum x)^2 = 50$

m = 105000/50 = 2100

For y-intercept b:

$\rm b = \dfrac{\sum y -m \sum x}{n}$

After calculating:

b = -4166040

The line becomes:

y = 2100x -4166040

If x = 2025 put this value in the line, we get:

y = 86460

Thus, the line of the best fit is y = 2100x -4166040 and the number of bottles is, 86460 which expect to sell in 2025

Learn more about the line of best fit here:

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2 years ago

Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A and an echelon form of A b

Rainbow [258]

Answer:

skip counting by 0

Step-by-step explanation:

skipcount by 0 to get to 100 for the third column.

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4 years ago

Convert the numeral to a numeral in base ten 24^five

seraphim [82]

Answer:

14 base 10

Step-by-step explanation:

Remember what the digits mean in the base 10 system we use.

For example, the number 539 (base 10) means:

The 5 in the hundreds place means 5 × 100 = 500.

The 3 in the tens place means 3 × 10 = 30.

The 9 in the ones place means 9 × 1 = 9.

The number 539 means

5 × 100 + 3 × 10 + 9 × 1 = 500 + 30 + 9 = 539

Also keep in mind one more thing. The ones digit means the number of ones. 1 = 10^0. The number in the ones digit means the number if 10^0 you have.

The ten digits is the digit of 10^1.

The hundreds digit is the digit of 10^2 since 10^2 = 100.

In other words, the ones digit is the number of 10^0 you have.

The tens digit is the number of 10^1 you have.

The hundreds digit is the number of 10^2 you have.

Now let's look at base 5.

Here, each place to the left is a successively higher power of 5 instead of a power of 10.

Think of the number 432 base 5.

The 2 is in the units place. it represents the number of 5^0 or ones you have. In this case, it is 2 × 5^0 = 2 × 1 = 2

The 3 is in the 5^1 place. It represents the number of fives you have.

3 × 5 = 15

The 4 is in the 5^2 place. It represents the number of 25's you have.

4 × 25 = 100

The number 432 base 5 means:

4 × 5^2 + 3 × 5^1 + 2 × 5^0

= 4 × 25 + 3 × 5 + 2 × 1

= 100 + 15 + 2

= 117 base 10

In other words, 432 base 5 = 117 base 10.

Now we do the same with your number, 24 base 5.

24 base 5 =

= 2 × 5^1 + 4 × 5^0

= 2 × 5 + 4 × 1

= 10 + 4

= 14

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1 year ago

The publishers of a business magazine are running sales promotions for the weekly magazine. The number of prospective customers

deff fn [24]

Answer:

the maximum expected number of subscriptions per week is

34 subscriptions

the minimum expected value of the extra income per week is

$94.20

Step-by-step explanation:

number of sales visits per day 1 - 40

success rate in closing sales 20%

total visits per week:

week 1 = 20 + 22 + 27 + 17 + 31 + 12 + 39 = 168
week 2 = 26 + 13 + 30 + 18 + 24 + 14 + 32 = 157
week 3 = 21 + 12 + 22 + 37 + 30 + 23 + 18 = 163
week 4 = 15 + 33 + 10 + 28 + 34 + 24 + 22 = 166
week 5 = 11 + 33 + 21 + 32 + 26 + 19 + 22 = 164
week 6 = 19 + 27 + 20 + 18 + 31 + 14 + 37 = 166
week 7 = 29 + 22 + 27 + 30 + 16 + 9 + 36 = 159
week 8 = 8 + 28 + 19 + 28 + 25 + 36 + 26 = 170

estimated sales per week:

week 1 = 168 x 20% = 33.6
week 2 = 157 x 20% = 31.4
week 3 = 163 x 20% = 32.6
week 4 = 166 x 20% = 33.2
week 5 = 164 x 20% = 32.8
week 6 = 166 x 20% = 33..2
week 7 = 159 x 20% = 31.8
week 8 = 170 x 20% = 34

extra income per week:

week 1 = 33.6 x $3 = $100.80
week 2 = 31.4 x $3 = $94.20
week 3 = 32.6 x $3 = $97.80
week 4 = 33.2 x $3 = $99.60
week 5 = 32.8 x $3 = $98.40
week 6 = 33.2 x $3 = $99.60
week 7 = 31.8 x $3 = $95.40
week 8 = 34 x $3 = $102

5 0

4 years ago

(6x-5y+4)dy+(y-2x-1)dx=0​

(6x-5y+4)dy+(y-2x-1)dx=0