Mathematics College

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Answers

Answer 1

(3.1) … … …

[tex]\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{2x-y}{x-2y}[/tex]

Multiply the right side by x/x :

[tex]\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{2-\dfrac yx}{1-\dfrac{2y}x}[/tex]

Substitute y(x) = x v(x), so that dy/dx = x dv/dx + v :

[tex]x\dfrac{\mathrm dv}{\mathrm dx} + v = \dfrac{2-v}{1-2v}[/tex]

This DE is now separable. With some simplification, you get

[tex]x\dfrac{\mathrm dv}{\mathrm dx} = \dfrac{2-2v+2v^2}{1-2v}[/tex]

[tex]\dfrac{1-2v}{2-2v+2v^2}\,\mathrm dv = \dfrac{\mathrm dx}x[/tex]

Now you're ready to integrate both sides (on the left, the denominator makes for a smooth substitution), which gives

[tex]-\dfrac12\ln\left|2v^2-2v+2\right| = \ln|x| + C[/tex]

Solve for v, then for y (or leave the solution in implicit form):

[tex]\ln\left|2v^2-2v+2\right| = -2\ln|x| + C[/tex]

[tex]\ln(2) + \ln\left|v^2-v+1\right| = \ln\left(\dfrac1{x^2}\right) + C[/tex]

[tex]\ln\left|v^2-v+1\right| = \ln\left(\dfrac1{x^2}\right) + C[/tex]

[tex]v^2-v+1 = e^{\ln\left(1/x^2\right)+C}[/tex]

[tex]v^2-v+1 = \dfrac C{x^2}[/tex]

[tex]\boxed{\left(\dfrac yx\right)^2 - \dfrac yx+1 = \dfrac C{x^2}}[/tex]

(3.2) … … …

[tex]y' + \dfrac yx = \dfrac{y^{-3/4}}{x^4}[/tex]

It may help to recognize this as a Bernoulli equation. Multiply both sides by [tex]y^{\frac34}[/tex] :

[tex]y^{3/4}y' + \dfrac{y^{7/4}}x = \dfrac1{x^4}[/tex]

Substitute [tex]z(x)=y(x)^{\frac74}[/tex], so that [tex]z' = \frac74 y^{3/4}y'[/tex]. Then you get a linear equation in z, which I write here in standard form:

[tex]\dfrac47 z' + \dfrac zx = \dfrac1{x^4} \implies z' + \dfrac7{4x}z=\dfrac7{4x^4}[/tex]

Multiply both sides by an integrating factor, [tex]x^{\frac74}[/tex], which gives

[tex]x^{7/4}z'+\dfrac74 x^{3/4}z = \dfrac74 x^{-9/4}[/tex]

and lets us condense the left side into the derivative of a product,

[tex]\left(x^{7/4}z\right)' = \dfrac74 x^{-9/4}[/tex]

Integrate both sides:

[tex]x^{7/4}z=\dfrac74\left(-\dfrac45\right) x^{-5/4}+C[/tex]

[tex]z=-\dfrac75 x^{-3} + Cx^{-7/4}[/tex]

Solve in terms of y :

[tex]y^{4/7}=-\dfrac7{5x^3} + \dfrac C{x^{7/4}}[/tex]

[tex]\boxed{y=\left(\dfrac C{x^{7/4}} - \dfrac7{5x^3}\right)^{7/4}}[/tex]

(3.3) … … …

[tex](\cos(x) - 2xy)\,\mathrm dx + \left(e^y-x^2\right)\,\mathrm dy = 0[/tex]

This DE is exact, since

[tex]\dfrac{\partial(-2xy)}{\partial y} = -2x[/tex]

[tex]\dfrac{\partial\left(e^y-x^2\right)}{\partial x} = -2x[/tex]

are the same. Then the general solution is a function f(x, y) = C, such that

[tex]\dfrac{\partial f}{\partial x}=\cos(x)-2xy[/tex]

[tex]\dfrac{\partial f}{\partial y} = e^y-x^2[/tex]

Integrating both sides of the first equation with respect to x gives

[tex]f(x,y) = \sin(x) - x^2y + g(y)[/tex]

Differentiating this result with respect to y then gives

[tex]-x^2 + \dfrac{\mathrm dg}{\mathrm dy} = e^y - x^2[/tex]

[tex]\implies\dfrac{\mathrm dg}{\mathrm dy} = e^y \implies g(y) = e^y + C[/tex]

Then the general solution is

[tex]\sin(x) - x^2y + e^y = C[/tex]

Given that y (1) = 4, we find

[tex]C = \sin(1) - 4 + e^4[/tex]

so that the particular solution is

[tex]\boxed{\sin(x) - x^2y + e^y = \sin(1) - 4 + e^4}[/tex]

Related Questions

Find the measure of the missing angles.

Answers

Answer:

b = 53

c = 53

Step-by-step explanation:

Alright so we already know that b and c are going to have the same angle measures. We can find b by subtracting 180 degrees to 127. Why you may ask? Its because when b and 127 are added together its obvious that it creates a straight line (supplementary angle). This means that two angles will sum up to 180 degrees. We can create an easy equation and solve for b.

127 + b = 180

b = 53 and c = 53

Best of Luck!

Kế hoạch đi dã ngoại của một gia đình sẽ bị hủy nếu trời có mây hoặc mưa. Biết xác suất để trời có mây là có mưa là có cả mây và mưa là . Tính xác suất để kế hoạch được thực hiện.

Answers

Answer:

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g is a trigonometric function of the form g(x)=acos(bx+c)+d
Below is the graph of g(x). The function has a maximum point at (3.5,-4) and a minimum point at (-1,-5)
Find a formula for g(x). Give an exact expression.

Answers

9514 1404 393

Answer:

a = 0.5b = π/4.5c = 7π/9d = -4.5

Step-by-step explanation:

The value of d is the average of the maximum and the minimum.

d = (-4 +-5)/2 = -9/2 = -4.5

The value of 'a' is the difference between the maximum and d.

a = -4 -(-4.5) = 0.5

The value of b is 2π divided by the period. Here, the half-period is 3.5 -(-1) = 4.5, so ...

b = 2π/9 = π/4.5

The value of c is the value that makes the cosine argument zero at x=3.5.

(π/4.5)(3.5) +c = 0

c = -7π/9

Raul is a manager at a local restaurant. He earns $18.50 per hour. How many hours per week does Raul
work if he earns $740 per week?

Answers

Answer: 40 hours
Just divide the total amount (740) by the rate per hour (18.50) and you get 40

For each of the indicates values given for x and y, determine which expression has a greater value (x+y)^2 or (x-y)^2

Answers

We can’t see the indicate values, but for a general answer, the first will be better AS LONG AS Y IS A POSITIVE. The second answer will be better if Y is a negative. Hope this helps.

Terrell loves to listen to music, so he buys a subscription to a music-streaming service. He pays $4.99 each month. How much does the streaming service cost per year?

Answers

Answer:
$59.88

Solving Steps:
So $4.99 each month. There are 12 months in a year so you have to times $4.99x12. This is $59.88 which is the answer.

find the exact value of tan(165°) using a difference of two angles

Answers

Answer: [tex]-2+\sqrt{3}[/tex]

=========================================================

Work Shown:

Apply the following trig identity

[tex]\tan(A - B) = \frac{\tan(A)-\tan(B)}{1+\tan(A)*\tan(B)}\\\\\tan(225 - 60) = \frac{\tan(225)-\tan(60)}{1+\tan(225)*\tan(60)}\\\\\tan(165) = \frac{1-\sqrt{3}}{1+1*\sqrt{3}}\\\\\tan(165) = \frac{1-\sqrt{3}}{1+\sqrt{3}}\\\\[/tex]

Now let's rationalize the denominator

[tex]\tan(165) = \frac{1-\sqrt{3}}{1+\sqrt{3}}\\\\\tan(165) = \frac{(1-\sqrt{3})(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}\\\\\tan(165) = \frac{(1-\sqrt{3})^2}{(1)^2-(\sqrt{3})^2}\\\\\tan(165) = \frac{(1)^2-2*1*\sqrt{3}+(\sqrt{3})^2}{(1)^2-(\sqrt{3})^2}\\\\\tan(165) = \frac{1-2\sqrt{3}+3}{1-3}\\\\\tan(165) = \frac{4-2\sqrt{3}}{-2}\\\\\tan(165) = -2+\sqrt{3}\\\\[/tex]

----------------------

As confirmation, you can use the idea that if x = y, then x-y = 0. We'll have x = tan(165) and y = -2+sqrt(3). When computing x-y, your calculator should get fairly close to 0, if not get 0 itself.

Or you can note how

[tex]\tan(165) \approx -0.267949\\\\-2+\sqrt{3} \approx -0.267949[/tex]

which helps us see that they are the same thing.

Further confirmation comes from WolframAlpha (see attached image). They decided to write the answer as [tex]\sqrt{3}-2[/tex] but it's the same as above.

What is the volume of the cylinder below?

Height 4
Radius 7

Answers

Answer:

V ≈ 615.75

r Radius 7

h Height 4

For the given annual rate of change, find the corresponding growth or decay factor. + 300%

Answers

Answer:

50% growth would would be (1 + .5)^ n

and 100% growth would be (1+ 1)^n

I assume that the answer would be (1+3)^n

for 300% growth the factor would be 3

Step-by-step explanation:

Diego made the shape on the left and Elena made the shape on the right. Each shape uses 5 circles.

Answers

Answer:

Step-by-step explanation:

Write the polynomial in standard form. Then name the polynomial based on its degree and number of
terms.
y-7y3 + 15y9

Answers

Answer:

[tex]15y^9 - 7y^3 + y[/tex]

Nonic polynomial

Step-by-step explanation:

Given

[tex]y - 7y^3 + 15y^9[/tex]

Required

Write in standard form

The standard form of a polynomial is:

[tex]ay^n + by^{n-1} + ......... + k[/tex]

So, we have:

[tex]y - 7y^3 + 15y^9[/tex]

The standard form is:

[tex]15y^9 - 7y^3 + y[/tex]

And the name is: Nonic polynomial (because it has a degree of 9)

Chad has a win loss ratio 5:5 across his games what percentage of games did he win

Answers

Answer:

Step-by-step explanation:

50%

Answer:

50%

Step-by-step explanation:

Win % = wins / total * 100%

Win% = (5/(5 + 5)) * 100

Win% = 5/10 * 100 = 50%

relationship between the two number
b 70.908
7.908
Which one is greater

Answers

Answer:

70.908 is greater than 7.908 .

I hope your day goes nice

Answer:

70.9 0 8 is Greater

both are different because of different placement of point.

What is a placement form from the school mean

Answers

Answer:

Placements are basically extended internships or work experience assignments

Step-by-step explanation:

PLEASE HELP

Solve the equation for y. Identify the slope and y-intercept then graph the equation.

-3y=3x-9

Y=
M=
B=

Please Include a picture of the graph and show your work if you can

Answers

9514 1404 393

Answer:

y = -x +3

m = -1

b = 3

Step-by-step explanation:

To solve the given equation for y, divide by the coefficient of y.

(-3y)/(-3) = (3x -9)/(-3)

y = -x +3

The slope is the x-coefficient, M = -1.

The y-intercept is the added constant, B = 3.

Both equations are graphed in the attachment. Texture has been added to the original so you can see the graphs are the same line.

find the area of the circle use 3.14 for pi d=4 m

Answers

Answer:

The area of circle is 12.56m sq.

How to find Joint and Combined variation?

Answers

Answer:

W is multiplied by 8

Step-by-step explanation:

If W varies jointly with x, y, and z, we can say that

W = k (xyz), with k being a constant for our original equation. We are asked what will happen to W if x, y, and z are each doubled. To figure this out, we can go back to our equation,

W = k (xyz)

First, we can double x, meaning that we multiply it by 2. Doing this, we get

W = k (2x * y * z)

Then, we can double y and z in a similar fashion, resulting in

W = k (2x * 2y * 2z)

W = 8 * k (xyz)

The new W, after all the doubling, is equal to 8 * k * x * y * z. The old W is equal to k * x * y * z. It can be determined that the new W is equal to 8 * the old W, so W is multiplied by 8

The firm had 15 billion VND of earnings before interest and tax (EBIT), corporate tax is 20%. The market price of stock is 60.000đ. Knowing that net income will be held 40% before using it for dividend. How much of the net income can be divided for shareholders?