1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
ra1l [238]
3 years ago
12

Find the solution of the differential equation that satisfies the given initial condition. y' tan x = 3a + y, y(π/3) = 3a, 0 &lt

; x < π/2, where a is a constant.
Mathematics
1 answer:
Paladinen [302]3 years ago
3 0

Answer:

y(x)=4a\sqrt{3}* sin(x)-3a

Step-by-step explanation:

We have a separable equation, first let's rewrite the equation as:

\frac{dy(x)}{dx} =\frac{3a+y}{tan(x)}

But:

\frac{1}{tan(x)} =cot(x)

So:

\frac{dy(x)}{dx} =cot(x)*(3a+y)

Multiplying both sides by dx and dividing both sides by 3a+y:

\frac{dy}{3a+y} =cot(x)dx

Integrating both sides:

\int\ \frac{dy}{3a+y} =\int\cot(x) \, dx

Evaluating the integrals:

log(3a+y)=log(sin(x))+C_1

Where C1 is an arbitrary constant.

Solving for y:

y(x)=-3a+e^{C_1} sin(x)

e^{C_1} =constant

So:

y(x)=C_1*sin(x)-3a

Finally, let's evaluate the initial condition in order to find C1:

y(\frac{\pi}{3} )=3a=C_1*sin(\frac{\pi}{3})-3a\\ 3a=C_1*\frac{\sqrt{3} }{2} -3a

Solving for C1:

C_1=4a\sqrt{3}

Therefore:

y(x)=4a\sqrt{3}* sin(x)-3a

You might be interested in
What is the equivalent expression of y+2/5y
Nonamiya [84]
The answer Is going to be 24
8 0
3 years ago
Find the surface area. SHOW WORK
anastassius [24]
22x12x5 and you will get the answer
4 0
3 years ago
4 coffees and 12 lattes. Costs $61
Gala2k [10]

Answer:

Cost of a coffee is <u>$2.5</u> and cost of a latte is <u>$4.25.</u>

Step-by-step explanation:

Let cost of 1 coffee be 'c' and cost of 1 latte be 'l' dollars.

Given:

4 coffees and 12 lattes cost $61.

12 coffees and 7 lattes cost $59.75.

∵ 1 coffee cost = c

∴ 4 coffees cost = 4c and 12 coffee cost = 12c

∵ 1 latte cost = l

∴ 12 lattes cost = 12l and 7 lattes cost = 7l

Now, as per question:

4c+12l=61-----1\\12c+7l=59.75----2

Now, multiplying equation (1) by -3 and adding the result to equation (2). This gives,

-3(4c+12l)=-3(61)\\\\-12c-36l=-183\\12c+7l=59.75\\+\\----------\\0-29l=-123.25\\\\l=\frac{-123.25}{-29}=\$4.25

Now, plug in the value of 'l' in equation 1 to solve for 'c'. This gives,

4c+12(4.25)=61\\\\4c+51=61\\\\4c=61-51\\\\4c=10\\\\c=\frac{10}{4}=\$2.5

Therefore, cost of a coffee is $2.5 and cost of a latte is $4.25.

4 0
3 years ago
Colin always takes his laundry to the laundromat for cleaning. Today, he has 5 items for dry cleaning and 12 items for regular w
kari74 [83]

Answer: 7 1/2 pounds

Step-by-step explanation:

7 0
2 years ago
WILL SOMEONE PLZ HELP ME!! I'VE ASKED LIKE 5 TIMES NOW!!! Approximately how long will it take you to double your savings of $500
USPshnik [31]

Answer:

18 years

Step-by-step explanation:

Rule of 72

(72)/(interest)=years to double

72/4=18

4 0
2 years ago
Other questions:
  • This circle centered at the origin and the Length of its radius is 8. What is the equation of the circle?
    10·2 answers
  • Please show work<br>its due soon<br>i really need answers
    14·1 answer
  • Suppose y varies directly with x. if y=6 when x=2 find x when y=12
    10·2 answers
  • And express delivery company charges by the The pound a package that weighs 5 pounds cost $24.50 overnight Ryan job equation to
    14·1 answer
  • Which rational number also belongs to the set of natural numbers? Select one: a. 15 b. -12 c. 38 d. 0.1666...
    13·1 answer
  • <img src="https://tex.z-dn.net/?f=%20x%20-%202%5Csqrt%7Bx%20-%203%20%3D%200%7D%20" id="TexFormula1" title=" x - 2\sqrt{x - 3 = 0
    14·1 answer
  • Carol has a savings account that earns .2 % interest annually. If the balance in her account is $6050.00, how much interest woul
    9·1 answer
  • Waldo has saved $14.55 in Nichols in quarters he has 21 more quarters and nickels how many quarters does Waldo have
    10·1 answer
  • Which are the numbers that are common multiples of 6 and 8. 12 18 24 30 40 48
    13·1 answer
  • How would you put that in an equation?
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!