Find the general solution of y''' + 9y'' + y' − 111y = 0 if it is known that y1 = e−6x cos(x) is one solution.

Answers

Answer 1

Answer:

[tex]\mathbf{y(x) = \dfrac{(c_1 + c_2) \ cos (x)}{e^{6x} }+ \dfrac{i(c_1 -c_2) \ sin (x)}{e^{6x}}+ c_3e^{3\ x}}[/tex]

Step-by-step explanation:

The objective of this question is to solve:

[tex]\dfrac{dy(x)}{dx}+ 9\dfrac{d^2y(dx)}{dx^2}+\dfrac{d^3y(x)}{dx^3}-111y(x) = 0 :[/tex]

Suppose the general solution is proportional to [tex]e^{\lambda x}[/tex] for [tex]\lambda[/tex] is constant; Then:

Let's replace [tex]y(x) = e^{\lambda\ x}[/tex] into the above equation:

i.e.

[tex]\dfrac{d^3}{dx^3}(e ^{\lambda x} )+ 9 \dfrac{d^2}{dx^2}(e ^{\lambda x} ) + \dfrac{d}{dx}(e ^{\lambda x} )- 111 \ e ^{\lambda x} = 0[/tex]

To Replace:

[tex]\dfrac{d^3}{dx^3}(e ^{\lambda x} )[/tex] with [tex]\lambda^3 e ^{\lambda x }[/tex]

[tex]\dfrac{d^2}{dx^2}(e ^{\lambda x} ) \ with \ \lambda^2 e^{\lambda\ x}[/tex]

[tex]\dfrac{d}{dx}(e ^{\lambda x} ) \ with \ \lambda e ^{\lambda \ x}[/tex]

Thus;

[tex]\lambda^3 e ^{\lambda x }[/tex] + [tex]9 \lambda^2 e^{\lambda\ x}[/tex] + [tex]\lambda e ^{\lambda \ x}[/tex] - 111 [tex]e ^{\lambda \ x}[/tex] = 0

[tex]e ^{\lambda \ x} (\lambda ^3 + 9 \lambda ^2 + \lambda - 111 )= 0[/tex]

In as much as [tex]e^{ \lambda x}\neq 0[/tex] for any finite [tex]\lambda[/tex]; Then:

[tex]\lambda ^3 + 9 \lambda ^2 + 111 = 0[/tex]

By Factorization:

[tex](\lambda - 3) ( \lambda ^2 + 12 \lambda + 37) = 0[/tex]

[tex]\lambda = -6 + i \ or\ \lambda = -6 - i \ or \ \lambda = 3[/tex]

However;

The root [tex]\lambda = -6 \pm i[/tex] yield;

[tex]y_1 = (x) = c_1 e ^ {(-6+i)x}[/tex]

[tex]y_2 (x) = c_2e^{(-6-i)x}[/tex]

The root [tex]\lambda = 3[/tex] yield;

[tex]y_3(x) = c_3 e^{3x}[/tex]

The general solution is:

[tex]y(x) = y_1(x) + y_2(x) + y_3(x) = \dfrac{c_1}{c^{(6-i)}x}+\dfrac{c_2}{c^{(6+i)}x}+ c_3e^{3x}[/tex]

Using Euler's Identity ;

[tex]e^{\alpha+i \beta} = e^\alpha \ cos (\beta ) + i \ e^\alpha \ sin ( \beta)[/tex]

[tex]y(x) = c_1 ( \dfrac{cos (x) }{e^{6x}}+ \dfrac{i \ sin x }{e^{6x} }) + c_2 ( \dfrac{cos (x)}{e^{6x}}- \dfrac{-i \ sin (x)}{e^{6x}})+c_3 e^{3x}[/tex]

[tex]\mathbf{y(x) = \dfrac{(c_1 + c_2) \ cos (x)}{e^{6x} }+ \dfrac{i(c_1 -c_2) \ sin (x)}{e^{6x}}+ c_3e^{3\ x}}[/tex]


Related Questions

Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a
mean of 100 and a standard deviation of 15.
106 124
The area of the shaded region is (Round to four decimal places as needed.)

Answers

Answer:

The area of the shaded region is 0.2898.

Step-by-step explanation:

Let X represent the IQ scores of adults.

It is provided that [tex]X\sim N(100,15^{2})[/tex].

Compute the probability of an adult having an IQ between 106 and 124 as follows:

[tex]P(106<X<124)=P(\frac{106-100}{15}<\frac{X-\mu}{\sigma}<\frac{124-100}{15})[/tex]

                             [tex]=P(0.40<Z<1.60)\\=P(Z<1.60)-P(Z<0.40)\\=0.94520-0.65542\\=0.28978\\\approx 0.2898[/tex]

*Use a z-table.

Thus, the area of the shaded region is 0.2898.

NEED HELP WITH MATH! Will Give Brainliest! Image is below.

Answers

Answer:

2

Step-by-step explanation:

What is the probability of scoring a total of 6

Answers

what are the choices
can you explain some more please

Picture is it............

Answers

In this triangle the three interior angles are = ∠82° , ∠54° and ∠x .

Then ;

∠82° + ∠54° + ∠x = 180° ( under angle sum property which says that the sum of the three interior angles of a triangle is equal to 180° )

= 136 ° + x = 180

= x = 180 - 136

= x = 44°

∠x and ∠y will sum up to 180° as they are a linear pair .

= 44° + y = 180

= y = 180 - 44

= y = 136°

Since ∠y and ∠z are vertically opposite angles their angle measure will be equal . Which means ;

= ∠y = ∠z

= 136° = ∠z

= ∠z = 136°

Therefore :-

The value of x = 44°

Change 3.5 decimal into a percent

Answers

Like out of 100? Then it would be 3.5%

The answer you're looking for is: 350%

(Tell me if i'm wrong please)

x=4/9,y=1/2, and z=3/5 6wy-3z

Answers

Answer: The answer would be - 7/5 or (-0.46)

Plug in the numbers that each letter equals. So you get 6(4/9)(1/2) - 3(3/5). Then you just simplify.

A survey of 120 college students is conducted and the results displayed below show that 80 students have a smartphone, 65 have a tablet, and 10 have neither. Fill in the empty table cells in the two-way frequency table.

Smartphone No Smartphone Total
Tablet 65
No Tablet 10
Total 80 120
What is the probability (rounded to the nearest whole percent) that a randomly selected college student has a tablet but not a smartphone? (4 points)

a
25%

b
30%

c
54%

d
38%

Answers

Answer:

the answer is 54 %

Step-by-step explanation:

divide 120/65=0.54, then since you're looking for the percent you multiply it by 100, and the answer would be 54%

Maria makes money
by commission rates.
She gets 15% of
everything she sells. If
Maria sold $23,000
worth of items this
month, what is her
salary for the month?
Show your work on
the slide!

Answers

Answer:

Monthly salary = $3450

Step-by-step explanation:

In this problem, it is given that, Maria gets 15% of  everything she sells. If Maria sold $23,000  worth of items this  month, we need to find her salary for the month.

It means we need to find the 15% of $23,000 to find her salary. Let the salary be x.

ATQ,

[tex]x=15\%\ \text{of}\ 23000\\\\x=\dfrac{15}{100}\times 23000\\\\x=\$3450[/tex]

Hence, her salary for the month is $3450.

Salary of Maria for the month is $3,450

Given that;

Amount of item sold by Maria = $23,000

Percentage of commission = 15%

Find:

Salary of Maria for the month

Computation:

Salary of Maria for the month = Amount of item sold by Maria × Percentage of commission

Salary of Maria for the month = $23,000 × 15%

Salary of Maria for the month = $3,450

Learn more:

https://brainly.com/question/19193145?referrer=searchResults

Write the decimal represented by each shaded square.

Answers

Answer:

1

Step-by-step explanation:

because all of the squares are shaded so it would be 1 whole





Divide.
636-3
The quotient is
and the remainder is

Answers

Answer:

212

Step-by-step explanation:

3 goes into 6 = 2 times

3 goes into 3 = 1 time

3 goes into 6 = 2 times

2 → 1 → 2 = 212

Hope this helps .


P. S pls mark as brainliest

A florist must make 5 identical
bridesmaid bouquets for a wedding. The budget is
$160, and each bouquet must have 12 flowers. Roses
cost $2.50 each, lilies cost $4 each, and irises cost
$2 each. The florist wants twice as many roses as the
other two types of flowers combined.

Answers

Answer:

A. 40x + 10y + 10z = $160

B. 8 Roses, 2 lilies and 2 irises

C.

1. 20x + 5y + 5z = $80

2. 4x + y + z = $16

3. 8x + 2y + 2z = $32

Step-by-step explanation:

Cost for each flower = $160/5 = $32

So we have $32 for each bouquet consisting of 12 flowers each.

Roses = x = $2.50 each

lilies = y = $4 each

irises = z = $2 each

8x + 2y + 2z = $32

8($2.50) + 2($4) + 2($2) = $32

$20 + $8 + $4 = $32

$32 = $32

a. Maximum budget is $160

40x + 10y + 10z = $160

40($2.50) + 10($4) + 10($2) = $160

$100 + $40 + $20 = $160

$160 = $160

b. From above

8x + 2y + 2z = $32

8 Roses, 2 lilies and 2 irises

c. No. There are other solutions If total cost is not limited

1. 20x + 5y + 5z

20($2.50) + 5($4) + 5($2)

$50 + $20 + $10

= $80

2. 4x + y + z

4($2.50) + $4 + $2

$10 + $4 + $2

= $16

3. 8x + 2y + 2z

8($2.50) + 2($4) + 2($2)

$20 + $8 + $4

= $32

The quanities x and y are proportional Find the constant of proportionality (r) in the equation y= rx

CORRECT ANSWER = BRAINLEIST​

Answers

Answer:

5

Step-by-step explanation:

7*5=35

12*5=60

20*5=100

PLEASE HEELP
Match the inverse, converse, contrapositive, and biconditional with their sentences.

Conditional Statement: If two angles are congruent, then they have the same measure.

Question 1 options:

converse


biconditional


inverse


contrapositive

1.
If two angles have the same measure, then they are congruent.

2.
If two angles are not congruent, then they do not have the same measure.

3.
If two angles do not have the same measure, then they are not congruent.

4.
Two angles are congruent if and only if they have the same measure.

Answers

Answer:

First, let's define each case:

A conditional statement is:

If P then Q.

P = hypothesis

Q = conclussion.

Converse: If Q then P.

Biconditional: P if and only if Q.

Inverse: If not P, then not Q.

Contrapositive: If not Q, then not P.

Our statement is:

"If two angles are congruent, then they have the same measure."

P = two angles are congruent.

Q = they have the same measure.

Now let's look at the options:

1) If two angles have the same measure, then they are congruent.

or: if Q then P, this is converse.

2) If two angles are not congruent, then they do not have the same measure.

or: If not P, then not Q, this is inverse.

3) If two angles do not have the same measure, then they are not congruent.

or: If not Q, then not P, this is the contrapositive.

4) Two angles are congruent if and only if they have the same measure.

or: P if and only if Q, this is biconditional.

The manager of a restaurant determined that the odds against a customer ordering dessert are 11/12 . What is the probability of a customer ordering dessert?

Answers

Answer:

1 out of 12

Step-by-step explanation:

12-11=1

2/3 divided by ? = 6/8 ASAP

Answers

Answer: 1/72 in decimal form 0.01389

Step-by-step explanation:

Answer:

im pretty sure it is 8/9

Step-by-step explanation:

6/8 x 8/9 =2/3

ax+6=15 a is a negative, what must be true about x?

Answers

Answer:

x= 9/a

Step-by-step explanation:

Find the equation of the line passing through the points (-2,-4) and (7,23)

Answers

Answer:

y=−5.4x

Step-by-step explanation:

Slope -5.4

Intercept -14.80

Whats the function to this math prob

Answers

Answer:

some reason i cant see the image is this just me

Step-by-step explanation:

Y is the vertical distance. The top of the curve is at y = 4 and the bottom of the curve is at y = -5, so the function would be between y -5 and y 5 which is written as :

D. -5 <= y <= 5

PLSSSS HELLPPPP MEEEEEEE ASAPPPP I WIILL GIVE BRAINLIEST!!!!!!!!! PLS SHOW WORK STEP BY STEP

Answers

Answer:

y = -3x + 3

Step-by-step explanation:

Slope = -3

y-intercept = 3

Substitute values into slope intercept form :

y=mx + b

Where :

m= Slope

b = y-intercept

[tex]y = - 3x + 3[/tex]

Two angles form a linear pair. The measure of one angle is 6 less then the measure of the other angle. Find the measure of each angle

Answers

Answer: one angle would be 50 degrees while the other would be 130 degrees.

Step-by-step explanation:

Please help me!!! ASAP!

Answers

Option C

Olivia paid $ 15683.28 after 8 years

What is 86.929 rounded to the nearest tenth ?

Answers

Answer:

86.9

Step-by-step explanation:

This is because after the decimal point, it goes to the tenths place and then the thousands place, therefore, if you round it, it would be 86.9.

not sure need help ​

Answers

Answer:

0.0078 or 7.8*10^-3

Answer:

7.8 * 10 ^-3

Step-by-step explanation:

( 1.3* 10 ^-5) ( 6 * 10^2)

Multiply the numbers

1.3 * 6 = 7.8

Multiply the exponent terms, remembering we add the exponents, when we multiply

10 ^ -5 * 10 ^2 = 10 ^( -5+2) = 10 ^-3

Put them back together

7.8 * 10 ^-3

Since this is in scientific notation, we are don

Translate subtract 10 from r, then subtract 4 from the result

Answers

Answer:

reste 10 de r, luego reste 4

Step-by-step explanation:Lets say the result is x x is a letter driving from spain so x=Spanish so translating that sentence to spanish is easy

Answer:

-r+6

Step-by-step explanation:

Triangle ABC is shown below.
What is the length of line segment AC?
A
A
O 7
O 9
2x
3x-7
O 14
O 18
B
4x - 10
с

Answers

Answer:

the answer is 14 C

Step-by-step explanation:

just took the test?

A process is normally distributed and in control, with known mean and variance, and the usual three-sigma limits are used on the control chart, so that the probability of a single point plotting outside the control limits when the process is in control is 0.0027. Suppose that this chart is being used in phase I and the averages from a set of m samples or subgroups from this process are plotted on this chart. What is the probability that at least one of the averages will plot outside the control limits when m = 5? Repeat these calculations for the cases where m = 10, m = 20, m = 30, and m = 50.

Required:
Discuss the results that you have obtained.

Answers

Answer:

The solution to the issue is outlined in the following portion of the summary.

Step-by-step explanation:

The given value is:

p = 0.0027

When m = 5,

⇒ [tex]P (x \geq 1) = 1 - P (x = 0)[/tex]

                   [tex]=1 - 5C0 (0.0027)^0 (1 - 0.0027)^5[/tex]

                   [tex]= 1 - 0.9866[/tex]

                   [tex]=0.0134[/tex]

When m = 10,

⇒ [tex]P (x \geq 1) = 1 - P (x = 0)[/tex]

                   [tex]=1 - 10C0 (0.0027)^0 (1 - 0.0027)^{10}[/tex]

                   [tex]=1 - 0.9733[/tex]

                   [tex]= 0.0267[/tex]

When m = 20,

⇒ [tex]P (x \geq 1) = 1 - P (x = 0)[/tex]

                   [tex]=1 - 20C0 (0.0027)^0 (1 - 0.0027)^{20}[/tex]

                   [tex]=1 - 0.9474[/tex]

                   [tex]=0.0526[/tex]

When m = 30,

⇒ [tex]P (x \geq 1) = 1 - P (x = 0)[/tex]

                   [tex]=1 - 30C0 (0.0027)^0 (1 - 0.0027)^{30}[/tex]

                   [tex]=1 - 0.9221[/tex]

                   [tex]=0.0779[/tex]

When m = 50,

⇒ [tex]P (x \geq 1) = 1 - P (x = 0)[/tex]

                   [tex]= 1 - 50C0 (0.0027)^0 (1 - 0.0027)^{50}[/tex]

                   [tex]= 1 - 0.8736[/tex]

                   [tex]=0.1264[/tex]

15 men can dig a ditch in 10 days, how many day
Will 10men take working at the same rate​

Answers

Answer:

If 10 men dig a ditch in 12 days .

Total man-days required to dig the ditch

= 10 men × 12 days

= 120 man-days

how long would 15 men take to dig it?

No of days required to finish the job by 15 men

= 120 men-days / 15 men

= 8 days

Answer: 8 days will be required to finish the job by 15 men

Step-by-step explanation:

Hope this helps u

Crown me as brainliest:)

If the volleyball team washes 10 more cars, then they will have met their overall goal of
washing 60 cars to raise money for their season. Let c represent the number of cars that
the volleyball team has washed so far. Which of the following equations can be used to
solve for c?

Answers

Answer:

Step-by-step explanation:

60-10=50

they washed 50 cars.

Write 81% as a fraction. There is no need to simplify your answer.​

Answers

Answer:

81/100

Step-by-step explanation:

Answer:

81/100

Step-by-step explanation:

How do you solve this? Without going into anything to complicated as this should be Year 10 maths

Answers

Given : ABCD is a square with each side 5cm .

To Find : The area of the shaded region .

Solution : On observing the figure we can see two quadrants , quadrant ADC & quadrant ABC .

If we join A to C , then it will be common for triangles ADC & ABC . And they will be congruent by SSS congruence condition.

Therefore the area of both quadrants will also be equal . Now we can find area of quadrant as ;

[tex]\large\boxed{\red{\bf Area_{quadrant}=\dfrac{\pi r^2}{4}}}[/tex]

Here radius will be equal to 5cm .

⇒ Area = πr² / 4 .

⇒ Area = π (5cm)² / 4 .

⇒ Area = 22/7 × 25 × 4 cm².

Area = 19.64 cm² .

So , total area of both quadrants = 39.28 cm² .

Also , area of square will be :

[tex]\large\boxed{\bf{\red{Area_{square}=(side)^2}}}[/tex]

⇒ Area = 5cm × 5cm .

Area = 25 cm².

Now , subtract area of one quadrant from the area of square = 25cm² - 19.64 cm² = 5.36 cm².

Similarly area of other white region = 5.36cm² .

And the areas sum will be = 5.36cm² × 2 = 10.72cm² .

Now , from the figure it's clear that ,

⇒ Area of unshaded region + Area of shaded region = 25cm².

⇒ 10.72cm² + ar( Shaded region ) = 25cm².

⇒ ar ( Shaded region ) = 25cm² - 10.72cm².

ar ( Shaded region ) = 14.28 cm².

Hence the area of shaded region is 14.28 cm².

[tex]\large\boxed{\red{\bf Answer = 14.28cm^2}}[/tex]

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WHAT GRANDPA SAID TO MARTHA ABOUT THE JACKET?16. WHAT HAPPENED AT THE END OF THE STORY?17. NAME THE MAIN CHARACTER18. NAME THE ANTAGONIST19. NAME THE TYPES OF CONFLICT20. NAME THE AUTHOR (WILL GIVE BRAINLIEST) What are some ways to stay focused when doing online schooling? Consider the rule y = 5x + 7.a. How many tiles are in Figure 0?b. Which figure has 37 tiles?c. In the equation y = mx + b, what do the letters m and b represent? A Special EveningI was a server at a fancy restaurant for about four years. I was pretty good at waiting tables, so I made good tips. People didn't take much notice of me as I refilled drinks or cleared empty plates. They just kept talking. Other times, a guest asked me to help make his or her loved one feel special. I always enjoyed helping people celebrate important occasions. One evening, a young man brought his date to the restaurant. He treated her wellheld the door open for her, slid her chair out, and pushed her chair in. I couldn't tell if it was their first date or not. The man seemed very anxious, but the woman seemed at ease. When the woman left the table, the man frantically waved for my attention. I rushed over and asked if he needed something. "I'm going to propose to my girlfriend. I need your help. When you bring the dessert, put this ring on top of the cake. Then, I will get down on one knee. Please make sure that nothing happens to the ring. It cost a fortune," the man said. "Sure thing, man. I'll take good care of the ring. Good luck," I said as I put it in my pocket. I was happy when I saw the couple laughing and holding hands. I was pretty sure that the young woman would accept the proposal. Finally, it was time to bring the dessert. Eagerly, I went to the kitchen and placed the diamond ring on top of a slice of cheesecake. Then, I had a brilliant idea of how to make the presentation even more special. I went to the storage room to get a candle and a lighter. When I returned to the kitchen, the cheesecake with the diamond ring was gone. Just as I reached the front of the restaurant to see if another server had taken it, I heard, "Yes! Robert, I will marry you!" Anxiously, I looked to see who had accepted the marriage proposal. I saw a happy-looking middle-aged woman admiring the diamond ring on her finger and a stunned-looking middle-aged man with his mouth hanging open in disbelief. Then, I noticed the guest who had given me the ring. He looked horrified. His face was pale and his eyes were bulging from their sockets. I glanced at his girlfriend. She was smiling and watching the woman with the diamond ring. She had no idea that the ring had been meant for her. I wasn't quite sure what to do. I suppose that my flight or fight response kicked in. I untied my apron, ran out the door, and never went back.A. The narrator tells the story without becoming involved.B. The narrator shows that the events personally affect him.C. The narrator makes excuses about his lack of courage.