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3 months ago

1. X^4(dy/dx) +x^3y =- sec (xy)Integral by separation of variables?

Mathematics

1 answer:

PIT_PIT [12.4K]3 months ago

8 0

Answer:

Step-by-step explanation:

Considering the differential equation x^4(dy/dx) + x^3y = -sec(xy). We will solve it employing the method of separation of variables;

$x^{4} \frac{dy}{dx} +x^{3}y = -sec(xy)\\x^{3}(x\frac{dy}{dx} + y) = -sec(xy)\\let \ v=xy\\\frac{dv}{dx} = x\frac{dy}{dx} + y(implicit \ in\ nature)\\$

By substituting v and dv/dx into the previous equation, we acquire;

$x^{3}\frac{dv}{dx} = -secv$

We then separate the variables:

$-\frac{dv}{secv} = \frac{dx}{x^{3} }$

$-cosvdv = x^{-3}dx\\ integrating\ both\ sides\\-\int\limits {cosv} \, dv = \int\limits {x^{-3} } \, dx\\-sinv = \frac{x^{-2} }{-2} + C\\since\ v = xy\\-sinxy = \frac{x^{-2} }{-2} + C\\2sin(xy) = x^{-2} -2C\\2 sin(xy) = \frac{1}{x^{2} } -K (where\ K = 2C)\\$

The end expression provides the solution to the differential equation.

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Let's apply the Pythagorean theorem to determine whether a triangle could be formed.

Assuming we have two wooden dowels each measuring 8 inches, we can break one dowel in half, resulting in two pieces of 4 inches each.

Now we have three wooden dowels with lengths of 4 inches, 4 inches, and 8 inches.

According to the Pythagorean theorem:

a² + b² = c²
4² + 4² = 8²
16 + 16 = 64
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To form a triangle, the length of the cut wooden dowel needs to exceed that of the uncut dowel.

I also visually attempted this exercise on paper, cutting according to the assumed dimensions. No triangle can be constructed with these pieces.

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3 months ago

Read 2 more answers

Ben and Joel are raising money for their class trip by selling wrapping paper. Ben raised $43.50 by selling 12 rolls of solid pa

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Detailed solution:

The price of solid paper is $1.75 each

Printed paper is priced at $2.50 each

Information provided:

Ben's total is $43.50 (12 rolls of solid paper and 9 rolls of printed paper)

Joel's total is $51.50 (8 rolls of solid paper and 15 rolls of printed paper)

Let x represent solid paper

Let y represent printed paper

The equations are: 12x + 9y = 43.50

8x + 15y = 51.50

To find x:

8x + 15y = 51.50

Solving for x gives: 8x = 51.50 - 15y

Thus, x = (51.50 - 15y)/8

Next, substitute x:

12x + 9y = 43.50

Replacing x results in: 12(51.50 - 15y)/8 + 9y = 43.50

Which expands to: (618 - 180y)/8 + 9y = 43.50

Multiplying through by 8 yields: 618 - 180y + 72y = 348

Rearranging gives us: -108y = 348 - 618

This simplifies to: -108y = -270

Dividing by -108 results in: y = 2.50

Using the expression for x:

x = (51.50 - 15y)/8

Substituting y, we get: x = (51.50 - 15(2.5))/8

Thus, x = (51.50 - 37.50)/8

This simplifies to: x = 14/8

Final value is: x = 1.75

Verification: x = 1.75; y = 2.5

Substituting into the first equation: 12x + 9y = 43.50

12(1.75) + 9(2.5) = 43.50

Calculating gives: 21 + 22.50 = 43.50

This confirms: 43.50 = 43.50

Substituting into the second equation: 8x + 15y = 51.50

8(1.75) + 15(2.5) = 51.50

Calculating yields: 14 + 37.50 = 51.50

This also confirms: 51.50 = 51.50

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2 months ago

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Answer:

33.75

Step-by-step explanation:

The manufacturer makes an average of 450 televisions daily, with 0.25% being defective each day.

Hence

0.25% of 450 gives the count of defective televisions produced per day

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0.25% /100 x 450

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This indicates that about 1.125 televisions produced on a daily basis are defective.

To estimate defective televisions over 30 days, multiply the daily defective output by 30, since the manufacturer averages 450 daily

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1.125 x 30

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Approximately 33.75 televisions will be defective over 30 days

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1. X^4(dy/dx) +x^3y =- sec (xy)Integral by separation of variables? ​

1. X^4(dy/dx) +x^3y =- sec (xy)Integral by separation of variables?