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Komok [63]
2 years ago
14

Use separation of variables to solve dy dx − tan x = y2 tan x with y(0) = √3. Find the value of c in radians, not degrees

Mathematics
1 answer:
a_sh-v [17]2 years ago
3 0

Answer:

y(x)=tan(-log(cos(x))+\frac{\pi }{3} )

Step-by-step explanation:

Rewrite the equation as:

\frac{dy(x)}{dx}-tan(x)=y(x)^{2} *tan(x)

Isolating \frac{dy}{dx}

\frac{dy}{dx} =tan(x)+tan(x)*y^{2}

Factor:

\frac{dy}{dx} =tan(x)*(1+y^{2} )

Dividing both sides by (1+y^{2} ) and multiplying them by dx

\frac{dy}{1+y^{2} } =tan(x)dx

Integrate both sides:

\int\ \frac{dy}{1+y^{2} } = \int\ tan(x)  dx

Evaluate the integrals:

arctan(y)=-log(cos(x))+C_1

Solving for y:

y(x)=tan(-log(cos(x))+C_1)

Evaluating the initial condition:

y(0)=\sqrt{3} =tan(-log(cos(0))+C_1)=tan(-log(1)+C_1)=tan(0+C_1)

\sqrt{3} =tan(C_1)\\arctan(\sqrt{3} )=C_1\\60=C_1

Converting 60 degrees to radians:

60degrees*\frac{\pi }{180degrees} =\frac{\pi }{3}

Replacing C_1 in the diferential equation solution:

y(x)=tan(-log(cos(x))+\frac{\pi }{3} )

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Answer:

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Step-by-step explanation:

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4 0
3 years ago
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Need some help with the answer to the question
kicyunya [14]

Answer:

(- 4, 27 )

Step-by-step explanation:

Equate the right sides of both equations, that is

x² - 2x + 3 = - 2x + 19 ← subtract - 2x + 19 from both sides

x² - 16 = 0 ← in standard form

(x - 4)(x + 4) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 4 = 0 ⇒ x = 4

x + 4 = 0 ⇒ x = - 4

Substitute these values into f(x) = - 2x + 19

f(4) = - 2(4) + 19 = - 8 + 19 = 11 ⇒ (4, 11 )

f(- 4) = - 2(- 4) + 19 = 8 + 19 = 27 ⇒ (- 4, 27 )

8 0
2 years ago
Peter has built a gazebo, whose shape is a regular heptagon, with a side length of $1$ unit. He has also built a pathway around
Sholpan [36]

Answer:

Area of pathway = 10.375 unit²

Step-by-step explanation:

Given - Peter has built a gazebo, whose shape is a regular heptagon, with

             a side length of 1 unit. He has also built a pathway around the

             gazebo, of constant width 1 unit, as shown below.

To find -  Find the area of the pathway.

Proof -

Central angle = \frac{360}{7} = 51.43°

Now,

Central angle / 2 = \frac{360}{14} = 25.71°

And

height = \frac{1}{2}.\frac{1}{tan25.41} = 1.038

Now,

Inner area of gazebo = 7·\frac{1}{2}·1·(1.038) = 3.634 unit²

Outer area of gazebo = (\frac{2.038}{1.038})²·(3.364) = 14.009 unit²

∴ we get

Area of pathway = Outer area - Inner area

                            = 14.009 - 3.634

                            = 10.375 unit²

⇒Area of pathway = 10.375 unit²

5 0
3 years ago
The fountain in the pond at the public park near Kevin's house has a pump that recirculates 75 gallons of water in 1/4 of an hou
pantera1 [17]
75/10 * 4/1 = 300/1     300 gallons every hour
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3 years ago
On an architect's blueprint, 1 inch corresponds to 8 feet. Find the length of a wall represented by a line 1 1/8 inches long on
Keith_Richards [23]
The problem is simply asking as to convert the measurement from the smaller scale to the actual scale. For this, we simply multiply the factor to the smaller measurement. We do as follows:

Length = 1 1/8 x 8 = 9 ft

Hope this helps.
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3 years ago
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