Mathematics College

SOLVE. y''+3y'+2y=4e^x cos3x

Answers

Answer 1

Solve the homogeneous equation

[tex]y'' + 3y' + 2y = 0[/tex]

Its characteristic equation is

[tex]r^2 + 3r + 2 = (r + 1) (r + 2) = 0[/tex]

with roots at [tex]r=-1[/tex] and [tex]r=-2[/tex], hence the characteristic solution is

[tex]y_c = C_1 e^{-x} + C_2 e^{-2x}[/tex]

For the nonhomogeneous equation, I'll use variation of parameters. We're looking for a solution of the form

[tex]y = u_1 y_1 + u_2 y_2[/tex]

to the equation

[tex]y'' + a(x) y'' + b(x) y = f(x)[/tex]

such that

[tex]\displaystyle u_1 = - \int \frac{y_2f(x)}{W(y_1,y_2)} \, dx[/tex]

[tex]\displaystyle u_2 = \int \frac{y_1 f(x)}{W(y_1,y_2)} \, dx[/tex]

The Wronskian [tex]W(y_1,y_2)[/tex] of the two fundamental solutions [tex]y_1=e^{-x}[/tex] and [tex]y_2=e^{-2x}[/tex] is

[tex]W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ {y_1}' & {y_2}' \end{vmatrix} = -e^{-3x}[/tex]

Then we have

[tex]\displaystyle u_1 = - \int \frac{e^{-2x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = 4 \int e^{2x} \cos(3x) \, dx[/tex]

[tex]\displaystyle u_2 = \int \frac{e^{-x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = -4 \int e^{3x} \cos(3x) \, dx[/tex]

Recall Euler's identity,

[tex]e^{(a+bi)t} = e^{at} (\cos(bt) + i \sin(bt))[/tex]

Then we have the general antiderivative

[tex]\displaystyle \int e^{(a+bi)t} \, dt = \frac1{a+bi} e^{(a+bi)t} + C = \frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C[/tex]

Taking the real parts of both sides, we have

[tex]\displaystyle \mathrm{Re}\left\{\int e^{(a+bi)t} \, dt \right\} = \mathrm{Re}\left\{\frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C\right\} \\\\ \int\,\mathrm{Re}\left\{e^{(a+bi)t}\right\} \, dt = \frac{e^{at}}{a^2+b^2} \mathrm{Re}\left\{(a-bi)(\cos(bt) + i \sin(bt))\right\} + C \\\\ \int e^{at} \cos(bt) \, dt = \frac{e^{at}}{a^2+b^2} (a\cos(bt)+b\sin(bt)) + C[/tex]

so that

[tex]\displaystyle u_1 = 4 \int e^{2x} \cos(3x) \, dx = \frac{4e^{2x}}{13} (2\cos(3x) + 3 \sin(3x))[/tex]

and

[tex]\displaystyle u_2 = -4 \int e^{3x} \cos(3x) \, dx = -\frac{2e^{3x}}3 (\cos(3x) + \sin(3x))[/tex]

We've found

[tex]y = u_1 y_1 + u_2 y_2[/tex]

[tex]\displaystyle y = \frac{4e^x}{13} (2\cos(3x) + 3 \sin(3x)) - \frac{2e^x}3 (\cos(3x) + \sin(3x))[/tex]

[tex]\displaystyle y = \frac2{39} e^x (5\sin(3x) - \cos(3x))[/tex]

Then the general solution to the differential equation is

[tex]\boxed{y(x) = C_1 e^{-x} + C_2 e^{-2x} + \frac2{39} e^x (5\sin(3x) - \cos(3x))}[/tex]

Related Questions

Find the first derivative (x²+x-2)/(2x²-2).
how to find the answer using the quotient rule.

Answers

the first derivative of the given function is:

f'(x) = (2x²- 12x - 2)/( 2x² - 2)²

How to find the derivative?

The quotient rule says that, if f(x) = g(x)/h(x), then:

f'(x) = ( g'(x)*h(x) - g(x)*h'(x) )/h(x)²

In this case we know that:

g(x) = x² + x - 2

h(x) = 2x² - 2

The derivatives are:

g'(x) = 2x + 1

h'(x) = 4x

Replacing that we get:

f'(x) = ( g'(x)*h(x) - g(x)*h'(x) )/h(x)²

= ( (2x + 1)*(2x² - 2) - ( x² + x - 2)*(4x) )/( 2x² - 2)²

Expanding the denominator we get:

f'(x) = ( 4x³ - 4x + 2x² - 2 - 4x³ + 4x² - 8x)/( 2x² - 2)²

f'(x) = (2x²- 12x - 2)/( 2x² - 2)²

That is the first derivative of the given function.

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There are 6 faces on a standard die. How many faces are on 5 dice?

Answers

Answer: 30

Step-by-step explanation: just do 6*5, 6 faces for five die is thirty faces

There are 30 faces on 5 dice in total.

Given that a standard die has 6 faces on it, we need to determine the number of the faces on 5 dice.

A standard die has 6 faces. If you have 5 dice, each with 6 faces, you can calculate the total number of faces by multiplying the number of dice by the number of faces on each die:

Total number of faces = Number of dice × Number of faces on each die

Total number of faces = 5 dice × 6 faces/die

Total number of faces = 30 faces

So, there are 30 faces on 5 dice.

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question is in picture please help

Answers

Answer:

[tex] \sf {a}^{6} [/tex]

Step-by-step explanation:

Given expression

[tex] \sf \: {a}^{0} \: • \: \: {a}^{6} [/tex]

To find expression solution

power rule

[tex] \sf \: {a}^{m} \: {a}^{n} = {a}^{m + n} [/tex]

[tex] \sf {a}^{0} + {a}^{6} = {a}^{0 + 6} [/tex]

[tex] \sf \therefore \: {a}^{6} [/tex]

[tex] \: \Large \mathbb{SOLUTION:}[/tex]

[tex] \: \: \rm{ {a}^{0} \times {a}^{6} }[/tex]

[tex] \: \: \text{Calculate the power }[/tex]

[tex] \: \: \underline{\underline{\rm{{a}^{6} }}}[/tex]

Solve by using the quadratic formula. Express the solution set in exact simplest form. y^2-7y-9=0

Answers

[tex]~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{1}y^2\stackrel{\stackrel{b}{\downarrow }}{-7}y\stackrel{\stackrel{c}{\downarrow }}{-9}=0 \qquad \qquad y= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ y= \cfrac{ - (-7) \pm \sqrt { (-7)^2 -4(1)(-9)}}{2(1)} \implies y = \cfrac{ 7 \pm \sqrt { 49 +36}}{ 2 } \\\\\\ y= \cfrac{ 7 \pm \sqrt { 85 }}{ 2 }\implies y= \begin{cases} \frac{ 7 + \sqrt { 85 }}{ 2 }\\[1em] \frac{ 7 - \sqrt { 85 }}{ 2 } \end{cases}[/tex]

need help in algebra 2

Answers

Answer:(-2,2)

Step-by-step explanation:

((12-4)(3-2))^2 + square root of 64

Answers

Answer: 72

Step-by-step explanation: First step that I did was to subtract 12 and 4 to get 8. I then subtracted 3 and 2 to get 1. Then since 8 and 1 are in the parenthesis, I multiplied 8 and 1 to get 8. 8 to the 2nd power then comes out to 64. 64 adding the square root of 64 which is 8 then comes out to 64 plus 8 because the square root of 64 is 8. 64 plus 8 equals 72.

find 1/3 + 3/43 sums and differences

Answers

Answer:

52/129

Step-by-step explanation:

PLEASE HELP!!!
Solve this exercise using the fact that the sum of the measures of the three angles of a triangle is 180° In a triangle, the measures of the three angles are x, x + 7, and x - 13. What is the measure of each angle?

Answers

Answer:

The sides are 62, 69, and 49

Step-by-step explanation:

x + (x + 7) + (x - 13) = 180

3x - 6 = 180

3x = 186

x = 62

In 2019, Company A purchased a commercial building with 8 retail units for $4,200,000. In 2021, Company A decided to sell this building for $4,500,000. Interpret the change as a percentage. Round to the nearest hundredth.

Answers

The change as a percentage is 6.7 %

How to calculate the percentage change ?

The price of the building in the year 2021 is $4,200,00

The price of the building in 2021 is $4,500,000

Therefore the percentage change can be calculated as follows

= 4,500,000 - 4,200,000/4,500,000 × 100

= 300,000/4,500,000 × 100

= 0.066 × 100

= 6.7 %

Hence the percentage change is 6.7%

Answers

35 over 56 is a proper fractions because 35 is lesser than 56 so it's proper likewise 144 over 196

a population has mean of 0.1 and standard deviation of 2.1.what is the probability that the mean of a random sample of size 900 will be negative

Answers

Answer on Question #54875

–

Math

–

Statistics

and Probability

A normal population has a mean of 0.1 and standard deviation 2.1. Find the probability that the

mean of a sample of size 900 will be negative.

Solution

In accordance with the given problem, we have the following data:

and

900

. It is known that

푥

(

)

and

(

)

Olivia counted thirty blue cars, ten red cars, and forty white cars in the parking lot. What is the ratio of white cars to blue cars, according to Olivia’s count?

3 to 4

1:4

4:1

43

Answers

Answer: I say 4/3 the fraction so your answer is D

Explanation: Because its white cars to blue cars the problem is whats equivalent to the ratio 40 to 30, 4:1 is not the same as 40 to 30 neither

is 1:4 or 3 to 4 So its D. :)

-1/3y+4x=3 solve for y

Answers

Answer: y = 12x − 9

Step-by-step explanation:

Let's solve for y

-1/3y +4x = 3

Step 1: Add -4x to both sides.

4x + -1/3y + -4x = 3 + -4x
-1/3y = -4x+3

Step 2: Divide both sides by (-1)/3.

-1/3y / -1/3 = -4x+3 / -1/3
y = 12 - 9

1. Simplify
-y/3+4x=3
2. Regroup terms
4x-y/3=3
3. Subtract 4x from both sides
-y/3 =3-4x
4. Multiply both sides by 3
-y=(3-4x)•3
5. Regroup terms
-y=3(3-4x)
6. Multiply both sides by -1
y=-3(3-4x)

Using the parent function, f(x)

, graph the new function after the transformations listed below:

a) Reflect the graph about the x-axis.

b)Vertical compress by a factor of 13 .

c) Vertical shift up 2.

d) Horizontal shift right 3.

Answers

The graphs of the transformed functions are below, the colors are:

a) g(x) = -x^2 → green.

b) g(x) = x^2/13 → blue

c) g(x) = x^2 + 2 → orange

d) g(x) = (x - 3)^2 → purple.

How to graph the transformed functions?

Here we have the quadratic parent function:

f(x) = x^2

And we want to apply some transformations, and then graph them.

The transformations are:

a) "Reflect the graph about the x-axis."

This is written as:

g(x) = -f(x)

Replacing f(x) by the actual function:

g(x) = -x^2

b) "Vertical compress by a factor of 13"

This is written as:

g(x) = f(x)/13 = x^2/13

c) "Vertical shift up 2."

This is written as:

g(x) = f(x) +2 = x^2 + 2

d) "Horizontal shift right 3."

This is written as:

g(x) = f(x - 3) = (x - 3)^2

The graphs are below, the colors are:

a) g(x) = -x^2 → green.

b) g(x) = x^2/13 → blue

c) g(x) = x^2 + 2 → orange

d) g(x) = (x - 3)^2 → purple.

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Bennett is setting up a light display for a wedding with two 10-foot strands of lights. The lights will be attached to the top of a pole and to two stakes on the ground forming two right triangles. If the distance from the base of the pole to each stake is the same as the height of the pole, how tall, in feet, should the pole be? Round your answer to the nearest tenth.

Answers

The height of the pole is 7.1 feet.

Given that a light display for a wedding with two 10-foot strands of lights and top of a pole and to two stakes on the ground forming two right triangles.

The Pythagorean theorem, also known as the Pythagoras theorem, explains the relationship between the three sides of a right triangle.

The two sides of triangle are equal that is x and and the third side is 10.

Now, we will apply the pythagoras theorem, we get

B²+P²=H²

Here B=P+x and H=10

x²+x²=(10)²

2x²=100

Now, we will divide both sides with 2, we get

2x²/2=100/2

x²=50

Now, we will take square root on both sides, we get

√x²=√50

x=√50

x=7.07106781

The Round to the nearest tenth is x=7.1 feet.

Hence, the height of the pole when wedding with two 10-foot strands of lights and the lights will be attached to the top of a pole and to two stakes on the ground forming two right triangles is 7.1 feet.

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What are the possible values of x for the following functions? F(x)=2-x / x(x-1)

Answers

Answer:

[tex]x \in \mathbb{R}-\{0,1\}[/tex]

Step-by-step explanation:

The denominator cannot be 0 (since division by 0 is undefined), meaning x cannot be 0 or 1.

So, the domain is

[tex]x \in \mathbb{R}-\{0,1\}[/tex]

Tanisha mowed five lawns last week she mowed each one and 712 hour she wrote the same ones this week in 5/12 hour using her lawnmower how many times longer was tanisha’s Time to mow all the lawns last week and this week express your answer as an improper fraction

Answers

Tanisha takes 5/6 hours longer to mow all the lawns last week than this week.

Here, we are given that Tanisha mowed 5 lawns last week in 7/ 12 hours each.

Thus, the total time taken by her to mow 5 lawns = 5(7/12)

= 35/ 12 hours

Now, this week she mows the five lawns in 5/12 hours each

Thus, the total time taken by her = 5(5/12)

= 25/12 hours

Therefore, the difference between the time taken by her last week and this week will be-

35/ 12 - 25/ 12

= (35 - 25)/ 12

= 10/ 12

= 5/6

Thus, Tanisha takes 5/6 hours longer to mow all the lawns last week than this week.

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Pls answer report if bad answer

Answers

Answer:

Step-by-step explanation:

24 x 3/4 = 18.

during last year my car depreciated by 30%. If the value of my car is $25,000 today, then what was the value of my car one year ago?

Answers

The value of my car one year ago was $35,714.3

INTEREST FORMULA: The interest formula includes two types of interests - simple interest and compound interest. Interest is the cost a borrower pays a lender for a loan. This additional sum, or the interest, must be paid in addition to the loan itself. The compound interest formula and the simple interest formula are both discussed in the interest formula.

Present value of the car = A = $25,000

Depreciation rate = r = 30% per annum

Time = t = 1 year

Let the initial value be p

We know the formula, A = P(1 – r/100)^t

$25,000 = P [1 – 30/100] ^1

$25,000 = P (7/10)

P = (25,000 * 10) / 7

P = $35,714.3

Therefore, One year ago, my car's value was worth $35,714.3.

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Helppppp! Line A and Line B are parallel and Line P is a transversal. The measure of angle 3 is 52°.
Find the measures of the rest of the angles in the diagram.

Answers

Answer:

Angle 1: 28 degrees

Angle 2: 52 degrees

Angle 3: 52 degrees

Angle 4: 28degrees

Angle 5: 28degrees

Angle 6 : 52degrees

Angle 7 : 52degrees

Angle 8 : 28degrees

Sports regulations usually give the sizes and weights of most sports equipment over a range of values. For each type of sports equipment, the diameters and weights to the appropriate type of ball are given in the accompanying table.
A box contains 30 baseballs. Write a compound inequality that shows the range of the weight of the balls. Graph the inequality.

Type of Ball Range of Diameters Range of Weights Tennis ball 2 1/2 inches to 2 5/8 inches no less than 2 ounces, no greater than 2 1/6 ounces Baseball 2 7/8 inches to 3 inches 5 to 5 1/4 ounces Basketball (women's) no less than 9.07 inches, no greater than 9.23 inches no less than 20 ounces, no greater than 22 ounces Soccer ball (size 5) no less than 21.64 inches, no greater than 22.28 inches 14.46 ounces to 15.87 ounces

Answers

The compound inequalities for all the balls that are sports regulated are { (x,y) : 2(1/2) ≤ x ≤ 2(5/8) and 2 ≤y≤ 2(1/6) } for Tennis ball.

{ (x,y) : 2(7/8) ≤ x≤ 3 and 5≤ y ≤ 5(1/4) } for Baseball.

{ (x,y) : 9.07 ≤ x ≤ 9.23 and 20 ≤ y ≤ 22 } for women's Basket ball.

{ (x,y) : 21.64 ≤x ≤ 22.28 and 14.46 ≤ y ≤ 15.87 } for Scoccer Ball.

What is the correct way of pronouncing a set in roaster form?

Suppose we are given { (x,y) : 9.07 ≤ x ≤ 9.23 and 20 ≤ y ≤ 22 } this reads as The set of all x comma y such that x is between this range and y is between in this range.

Sports regulations have some rules that the diameters and weights of sports pieces of equipment must lie between the regulations that have been set.

We are denoting the Diameters of the balls as x in inches and the weights of the balls as y in ounces.

Now for a Tennis ball the range for x and y can be written in roaster form of set is { (x,y) : 2(1/2) ≤ x ≤ 2(5/8) and 2 ≤y≤ 2(1/6) }.

For Baseball it is { (x,y) : 2(7/8) ≤ x≤ 3 and 5≤ y ≤ 5(1/4) }.

For women's Basketball it is { (x,y) : 9.07 ≤ x ≤ 9.23 and 20 ≤ y ≤ 22 }.

For Soccer ball the set in roaster form can be written as

{ (x,y) : 21.64 ≤x ≤ 22.28 and 14.46 ≤ y ≤ 15.87 }.

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14. 5.2v-6-8.5v simplify

Answers

The answer of the given equation will be -6 - 3.3v

For doig the sum we need to first know what is algebric simplification

So Algebric Simplification is basically is writting the same algebric expression with no same term in a compact manner. For solving the simplification weed to first group the like terms and solve the brackets first if any then in that simplified equation we will be left with unlike variables that cannot be simplified further.

To make it more easier remember the rule of BODMAS it stands for Brackets Of Dvision Multiplication Addition Substraction in any sum where it will be asked to simplify any expression we will go by the order of the above BODMAS rule first we will solve the brackets after that divisions then multiplications followed bty addition and subtraction .

In the given sum as these r not given we will simply group the expressions with alike variable that is (5.2v - 8.5v) which will result in - 3.3v and we cannot to do any further calculations so the answer will be -6 - 3.3v

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Points A and B have coordinates A(-4, 2) and B(3,-6). Find the coordinates of point P, the weighted average of points A and B, in which point A
has a weight of 2 and point 8 has a weight of 5.
A) (-2,-²)
B) (-2)
C) (1,-)
D) (1,-13)

Answers

Answer:

C) P(1, -26/7)

Step-by-step explanation:

You want the weighted average of A(-4, 2) and B(3,-6) using weights 2 and 5, respectively.

Weighted average

Each of the values is multiplied by the corresponding weight, and the sum of those products is divided by the total of the weights to obtain the weighted average.

P = (2A +5B)/(2+5)

P = (2(-4, 2) +5(3, -6))/7 = (-8 +15, 4 -30)/7 = (7, -26)/7

P = (1, -26/7)

The coordinates of P are (1, -26/7).

maths geometry questions
pleas help

Answers

Step-by-step explanation:

remember, a parallelogram is a quadrilateral with 2 pairs of parallel and equal sides, and 2 pairs of correspondingly equal angles. and its diagonals are intersecting each other in their midpoint.

with that in mind

(i)

because ABCD is a parallelogram, AD = BC, and AB = CD.

and angle A = angle C (and angle B = angle D).

therefore, as AE = FC, this also means that ED = BF.

via SAS (side angle side) it is clear that the triangles ABE and CDF are congruent.

therefore, EB = DF.

as ED is part of AD, and BF is part of BC, it is also clear that ED || BF.

together with ED = BF this means that EB || DF.

so, we have with BEDF 2 pairs of parallel and equal sides. and that is a parallelogram.

(j)

BCDE and ABCG are parallelograms.

so, BC = DE = AG, CD = BE, AB = CG.

due to the midpoint intersection rule of the diagonals of parallelograms, we also know

GE = CG = AB, DG = BG.

and as CG || AB, then also GE || AB.

together with AB = GE that means that AE = BG and AE || BG.

so, we have with ABGE again 2 pairs of parallel and equal sides. and that is a parallelogram.

Last year, Salma invested her money in two purchases. She purchased a certificate of deposit for $3000 that paid 5% interest per year and purchased $7000 in corporate bonds paying 4% interest per year. Answer the questions below. Do not do any rounding.

Answers

Answer:

a. $2850

b. $6720

Step-by-step explanation:

3000(1-0.05) = $2850

7000(1-0.04) = $6720

ILL give BRAINIEST

What is the width of a rectangle with a length of 15 cm and area of 125 cm

No fake answers

Answers

Answer:

w≈8.33

Step-by-step explanation:

Greetings !

Firstly recall rectangle area formula

Thus,

[tex]area \: of \: rectangle \: = length \: \times \: width[/tex]

Given values:-

length = 15cmarea = 125cm

require value:-

width =?

solution/ work-out:-

[tex]width = \: \frac{area \: of \: rectangle}{length} [/tex]

[tex]w = \frac{125}{15} [/tex]

[tex]w = 8.333...[/tex]

Hope it helps !!!

5. Combien de paires de nombres naturels peut-on former dont : a) la somme est 23? b) la différence est 100?

Answers

Answer: a) 24 b) infini

Step-by-step explanation: a) parce que tu peut faire que 24 somme pour faire 23. B) infini car tout plus haut 100 fonctione

24. Knowing a Person Who Was Murdered In a sample of 11,771 children
ages 2 to 17, 8% have lost a friend or relative to murder. Four children are
selected at random. (Adapted from University of New Hampshire)
(a) Find the probability that all four have lost a friend or relative to murder.
(b) Find the probability that none of the four has lost a friend or relative to
(c) Find the probability that at least one of the four has lost a friend or
relative to murder.

Answers

Using the binomial distribution, the probabilities are given as follows:

a) 0.000041 = 0.0041%.

b) 0.7164 = 71.64%.

c) 0.2836 = 28.36%.

For each person, there are only two possible outcomes, either they have lost a friend, or they have not. The probability of a person having lost a friend is independent of any other person, hence the binomial distribution is used to solve this question.

What is the binomial distribution formula?

The formula for the probability of x successes is:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are given by:

n is the number of trials of the experiment.p is the probability of a success on a single trial of the experiment.

For this problem, the values of the parameters are:

n = 4, p = 0.08.

The probability of all four is P(X = 4), hence:

P(X = 4) = (0.08)^4 = 0.000041.

0.000041 = 0.0041% probability that all four have lost a friend or relative to murder.

The probability of none is P(X = 0), hence:

P(X = 0) = (0.92)^4 = 0.7164.

0.7164 = 71.64% probability that none have lost a friend or relative to murder.

The probability of at least one is given by:

P(X >= 1) = 1 - P(X = 0)

Then:

1 - 0.7164 = 0.2836.

0.2836 = 28.36% probability that at least one have lost a friend or relative to murder.

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the sum of two consecutive integers is 187. find the integers

Answers

Answer:

92.5 and 94.5

___________________

Solve

[tex]187 \div 2 = 93.5\\\mathrm{Split\:93.5\:into\:two\:integers\:by\pm1}\\93.5 - 1 = 92.5\\93.5 + 1 = 94.5[/tex]

→ Thus, we have our two integers.

_________________________

Simplify the quantity 9 minus two thirds times the square root of 9 end quantity squared plus the quantity 1 minus 6 end quantity squared.

74
24
495
1,225

Answers

The correct answer is option A that is 74.

First, we need to put the expression in the form of mathematical equation. A mathematical equation may be defined as the combination of numbers and arithmetic symbols which are related to each other according to the expression given. The expression in the form of mathematical expression can be written as:

(9 - 2/3×√9)² + (1-6)²

Solving the brackets separately.

(9 - 2/3×3)² + (-5)²

Now in the next step, simplify the equation.

=>(9 - 2)² + 25

=>(7)² + 25

=>49 + 25

=>74.

Therefore, the required answer is 74.

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