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
VashaNatasha [74]
3 years ago
14

Solve the following differential equations or initial value problems. In part (a), leave your answer in implicit form. For parts

(b) and (C), write your answer in explicit form.
a. y'= t²+7/y⁴-4y³
b. y'= (cos²y) ln t
c. (t²+t)y' +y² = ty²+, y(1)= -1
Mathematics
1 answer:
shepuryov [24]3 years ago
8 0

Answer:

(a) (y^5)/5 + y^4 = (t^3)/3 + 7t + C

(b) y = arctan(t(lnt - 1) + C)

(c) y = -1/ln|0.09(t + 1)²/t|

Step-by-step explanation:

(a) dy/dt = (t^2 + 7)/(y^4 - 4y^3)

Separate the variables

(y^4 - 4y^3)dy = (t^2 + 7)dt

Integrate both sides

(y^5)/5 + y^4 = (t^3)/3 + 7t + C

(b) dy/dt = (cos²y)lnt

Separate the variables

dy/cos²y = lnt dt

Integrate both sides

tany = t(lnt - 1) + C

y = arctan(t(lnt - 1) + C)

(c) (t² + t) dy/dt + y² = ty², y(1) = -1

(t² + t) dy/dt = ty² - y²

(t² + t) dy/dt = y²(t - 1)

(t² + t)/(t - 1)dy/dt = y²

Separating the variables

(t - 1)dt/(t² + t) = dy/y²

tdt/(t² + t) - dt/(t² + t) = dy/y²

dt/(t + 1) - dt/(t(t + 1)) = dy/y²

dt/(t + 1) - dt/t + dt/(t + 1) = dy/y²

Integrate both sides

ln(t + 1) - lnt + ln(t + 1) + lnC = -1/y

2ln(t + 1) - lnt + lnC = -1/y

ln|C(t + 1)²/t| = -1/y

y = -1/ln|C(t + 1)²/t|

Apply y(1) = -1

-1 = ln|C(1 + 1)²/1|

-1 = ln(4C)

4C = e^(-1)

C = (1/4)e^(-1) ≈ 0.09

y = -1/ln|0.09(t + 1)²/t|

You might be interested in
If (6y − 11)(6y + 11) = ay2 − b, what is the value of a?
lara31 [8.8K]
I hope this helps you

4 0
3 years ago
What is the equation of the line that passes through (0,3) and (-5,3)
ddd [48]

Answer:

y=+3

Step-by-step explanation:

the equation of a line is y=mx+c

where m is the slope and c is the y-intercept.

First, let's find what m is, the slope of the line...

The slope of a line is a measure of how fast the line "goes up" or "goes down". A large slope means the line goes up or down really fast (a very steep line). Small slopes means the line isn't very steep. A slope of zero means the line has no steepness at all; it is perfectly horizontal.

For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form:

So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (0,3), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=0 and y1=3.

Also, let's call the second point you gave, (-5,3), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=-5 and y2=3.

Now, just plug the numbers into the formula for m above, like this:

m=\frac{3-3}{-5-0}

or

m=\frac{0}{-5}

or

m=0

So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+c like this:

y=0x+c

Now, what about c, the y-intercept?

To find c, think about what your (x,y) points mean:

(0,3). When x of the line is 0, y of the line must be 3.

(-5,3). When x of the line is -5, y of the line must be 3.

Because you said the line passes through each one of these two points, right?

Now, look at our line's equation so far: y=0x+c. c is what we want, the 0 is already set and x and y are just two "free variables" sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the two points (0,3) and (-5,3).

So, why not plug in for x and y from one of our (x,y) points that we know the line passes through? This will allow us to solve for c for the particular line that passes through the two points you gave!.

You can use either (x,y) point you want..the answer will be the same:

(0,3). y=mx+c or 3=0x 0+c, or solving for c: c=3-(0)(0). c=3.

(-5,3). y=mx+c or 3=0x -5+c, or solving for c: =3-(0)(-5). c=3.

See! In both cases we got the same value for c. And this completes our problem.

The equation of the line that passes through the points

(0,3) and (-5,3)  

is  

y=+3

Hope this helps :)

7 0
3 years ago
Read 2 more answers
Nguyen is decorating the room for her sister’s baby shower party. She is wrapping Vietnamese conical hats in baby prints tissue
kodGreya [7K]

Answer:

She needs 301.8 in² of tissue paper.

Step-by-step explanation:

In order to calculate the amount of tissue paper she needs to make 21 hats, we need to find the area of each hat and multiply it by the total number of hats she wants to make.

Since the Vietnamese conical hats are hollow and don't have a base, we need to find the area of the it's lateral walls, this is given by the following expression:

lateral area = pi*r*l

r = d/2 = 16/2 = 8 in

We were given the height, but we need the slant height, therefore:

l = sqrt(r² + h²) = sqrt(8² + 9²) = sqrt(64 + 81) = sqrt(145)  = 12.041 in

Then the lateral area is:

lateral area = pi*8*12.0141 = 301.8 in²

She needs 301.8 in² of tissue paper.

4 0
3 years ago
2/x + 3/y = 13 ; 5/x - 4/y = -2​
Tomtit [17]

Given that

(2/x)+(3/y) = 13--------(1)

(5/x)-(4/y) = -2 -------(2)

Put 1/x = a and 1/y = b then

2a + 3b = 13 ----------(3)

On multiplying with 5 then

10a +15 b = 65 -------(4)

and

5a -4b= -2 ----------(5)

On multiplying with 2 then

10 a - 8b = -4 -------(6)

On Subtracting (6) from (4) then

10a + 15b = 65

10a - 8b = -4

(-)

_____________

0 + 23 b = 69

______________

⇛ 23b = 69

⇛ b = 69/23

⇛ b =3

On Substituting the value of b in (5)

5a -4b= -2

⇛ 5a -4(3) = -2

⇛ 5a -12 = -2

⇛ 5a = -2+12

⇛ 5a = 10

⇛ a = 10/5

⇛ a = 2

Now we have

a = 2

⇛1/x = 2

⇛ x = 1/2

and

b = 3

⇛1/y = 3

⇛ y = 1/3

<u>Answer :-</u>The solution for the given problem is (1/2,1/3)

<u>Check</u>: If x = 1/2 and y = 1/3 then

LHS = (2/x)+(3/y)

= 2/(1/2)+3/(1/3)

= (2×2)+(3×3)

= 4+9

= 13

= RHS

LHS=RHS is true

and

LHS=(5/x)-(4/y)

⇛ 5/(1/2)- 4/(1/3)

⇛(5×2)-(4×3)

⇛ 10-12

⇛ -2

⇛RHS

LHS = RHS is true

5 0
2 years ago
Clara's suitcase weighs 58 1/4 pounds, but the airline's weight limit for suitcases is 50 pounds. How much weight does she need
sergij07 [2.7K]
8 pounds and 4 oz. 58.04 - 8.04=50
5 0
3 years ago
Other questions:
  • A machine make 24 items in 8 minutes how many can it make in 14
    13·1 answer
  • 24 is 60% of what number
    8·1 answer
  • Simplify 7 + (−3). plz help
    11·1 answer
  • Draw an array then write a fact family to describe your array
    9·2 answers
  • Find the x.intercept and the y.intercept of the graph of each quation. Then graph the equation
    11·1 answer
  • Mrs. Kelly bought a bag of flour that holds 1623 cups of flour. Each loaf of banana bread uses 23 cup of flour.
    5·2 answers
  • Mr. Evans has a plot of land in the shape of a right triangle that he wants to cover with grass sod. The diagram shows his plot
    9·1 answer
  • Which three-dimensional solid could be represented by this net?
    14·1 answer
  • The shelf life of a particular dairy product is normally distributed with a
    6·1 answer
  • an electrician charges an initial fee of $250 plus and hourly fee of $25. Mr.Scott paid the electrician $400. How many hours did
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!