11r-12r+15
combine like terms

It would take approximately 19.65 years for an investment of $100,000 at a 7% annual interest rate to grow to $500,000. Rounded to two decimal places, the answer is 19.65 years.

To determine the number of years it will take for an investment of $100,000 at a 7% annual interest rate to grow to $500,000, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the investment ($500,000)

P = the initial principal amount ($100,000)

r = the annual interest rate (7% or 0.07)

n = the number of times that interest is compounded per year (assuming annually, so n = 1)

t = the number of years

Plugging in the values we know, we get:

$500,000 = $100,000(1 + 0.07/1)^(1*t)

Dividing both sides of the equation by $100,000 and simplifying:

5 = (1.07)^t

To solve for t, we can take the logarithm of both sides:

log(5) = log[(1.07)^t]

Using logarithmic properties, we can bring down the exponent:

log(5) = t * log(1.07)

Finally, we can solve for t by dividing both sides by log(1.07):

t = log(5) / log(1.07)

Using a calculator, we find:

t ≈ 19.65

Therefore, it would take approximately 19.65 years for an investment of $100,000 at a 7% annual interest rate to grow to $500,000. Rounded to two decimal places, the answer is 19.65 years.

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Answer:

  k = -3

Step-by-step explanation:

The expression evaluates at x=3 to ...

  (2·3² +3k -9)/(3² -4·3 +3) = (9+3k)/0

In order for the limit to exist, there must be a "hole" at x=3. That will only be the case when ...

  9 +3k = 0

  k = -9/3 = -3

The value of k must be -3 for the limit to exist.

__

When k=-3, the expression factors as ...

  \(\dfrac{2x^2-3x-9}{x^2-4x+3}=\dfrac{(x-3)(2x+3)}{(x-3)(x-1)}=\dfrac{2x+3}{x-1}\quad x\ne3\)

The limit as x→3 is 9/2.

Answer: 49

Step-by-step explanation:

Given

The dimension of the block is \(24.5\ in.\times12.5\ in.\times 8\ in.\)

Each vase requires \(50\ in.^3\) of clay

The volume of clay block is

\(\Rightarrow V=24.5\times 12.5\times 8\\\Rightarrow V=2450\ in.^3\)

Number of vases that can be made

\(\Rightarrow n=\dfrac{2450}{50}\\\\\Rightarrow n=49\ \text{Vases}\)

Thus, 49 vases can be made

Answer:

15

Step-by-step explanation:

We have the hypotenuse and adjacent values of the sides in the triangle provided, therefore we will be using cosine.

\(cos(21.3)=\frac{14}{x}\)

Multiply both sides by x:

\(xcos(21.3)=14\)

Divide both sides by cos(21.3):

\(x=\frac{14}{cos(21.3)}\)

Solve using calculator:

15.0264

Round to nearest whole number:

15

Answer:

B. (-7,-3)

Step-by-step explanation:

Given the circle: \((x + 7)^2 + (y - 10)^2 = 13^2\)

The point (x,y) which lie on the circle are the coordinate which satisfies the given equation of the circle.

We now consider the given options.

Option A (5,12)

When x=5, y=12

\((5 + 7)^2 + (12 - 10)^2 =12^2+2^2=148\neq 169= 13^2\)

Option B (-7,-3)

When x=-7, y=-3

\((-7 + 7)^2 + (-3 - 10)^2 =0^2+(-13)^2= 169=13^2\)

Option C (-6,-10)

When x=-6, y=-10

\((-6 + 7)^2 + (-10 - 10)^2 =1^2+(-20)^2=401\neq 169= 13^2\)

Option D (6,23)

When x=6, y=23

\((6 + 7)^2 + (23 - 10)^2 =13^2+13^2=338\neq 169= 13^2\)

We can see that only (-7,-3) satisfies the equation of the circle. Thus it is the point which lies on the circle.

The correct option is B.

Answer: 80un^2

Step-by-step explanation:

The length of AB is 10

The length of BC is 8

The length of CD is 10

The length of DA is 8

10 x 8 = 80 and there are two pairs of 8 and ten which makes it 80un^2

Answer:

Joy will need to use 9 cups of strawberries for 6 cups of blueberries.

Step-by-step explanation:

Given,

Cups of blueberries used for 3 cups of strawberry = 2

Ratio of cups of blueberry to strawberry = 2:3

Joy wants to use 6 cups of blueberries.

Let,

x be the cups of strawberry for 6 cups of blueberries.

Ratio of cups of blueberries to strawberries = 6:x

Using proportion;

Ratio of cups of blueberry to strawberry :: Ratio of cups of blueberries to strawberries

Product of extreme = Product of mean

Dividing both sides by 2

Answer:

\(\pi432 {cm}^{3} \)

Step-by-step explanation:

The formula to calculate the volume of a cone is

\(v = \frac{(\pi {r}^{2})h }{3} \)

Since they asked that we determine the volume in terms of pi, we are not going to equate pi to its numerical value.

\(r = 9 \: \: \: \: \: \: \: \: h = 16\)

I am not sure if the radius is 9 cm because the image is kinda blurry. If it is not 9 cm you can input the actual radius into the formula to find the volume.

\(v = \frac{(\pi {r}^{2})h }{3} \\ v = \frac{(\pi {9}^{2} )16}{3} \\ v = \frac{(\pi81)16}{3} \)

\(v = \frac{\pi1296}{3} \\ v = \pi432\)

Answer:

x=45

Step-by-step explanation:

30/x = 54/81

30 = 54/81(x)

x = 30 ÷ 54/81

x=45

Answer:

x = 45

Step-by-step explanation:

30/x = 54/81

~Cross multiply

30 * 81 = 54 * x

~Simplify

2430 = 54x

~Divide 54 to both sides

45 = x

Best of Luck!

Answer:

Perimeter: 14a+2b

Tickets:7x+6

Step-by-step explanation:

Perimeter: Add all the functions together

Tickets: Subtract 15x+1 by 8x-5.

To find the average rate of change of a function over an interval, we need to calculate the slope of the secant line connecting the points on the function at the two endpoints of the interval.

The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept. In the case of a secant line, we can calculate the slope by using the formula:

m = (y2 - y1) / (x2 - x1)

For the function y = x^2 + 4x - 1, we can plug in the values of x and y at the two endpoints of the interval, (-4 and 1), to find the slope of the secant line:

m = [(1)^2 + 4(1) - 1] - [(-4)^2 + 4(-4) - 1] / (1 - (-4))

Simplifying this expression, we get:

m = (1 + 4 - 1) - (16 + 16 - 1) / (-3)

This simplifies to:

m = -20 / -3

Therefore, the average rate of change of the function y = x^2 + 4x - 1 over the interval -4 ≤ x ≤ 1 is 6.5.

The Determinant of the product of \(5B^T and  7B^T is  (35)^n * 9\).

We are given that A and B are n×n matrices with det A = 5 and det B = -3. We need to find the determinant of the product of \(5B^T\) and \(7B^T\) , which is \(det(5B^T)\) * \(det(7B^T)\).

First, let's find the determinant of \(5B^T\). We know that \(det(B^T)\) = det(B), and det(B) = -3.

Therefore, \(det(5B^T) = 5^n\) * \(det(B^T) = 5^n\) * (-3), where n is the dimension of the matrix.

Next, we find the determinant of \(7B^T\).

Similarly, \(det(7B^T) = 7^n * det(B^T) = 7^n * (-3)\).

Now, we multiply these determinants to find the determinant of the product:
\(det(5B^T) * det(7B^T) = (5^n * (-3)) * (7^n * (-3)) = (5^n * 7^n) * (-3)^2 = (5*7)^n * 9.\)

So, the determinant of the product of \(5B^T\) and \(7B^T\) is \((35)^n * 9\).

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43% of elementary students chose something other than blue, red, or yellow as their favorite color.

Blue: 24%

Red: 17%

Yellow: 16%

Total: 24% + 17% + 16% = 57%

Percentage chose something other:

100% - 57% = 43%.

Therefore, 43% of elementary students chose something other than blue, red, or yellow as their favorite color.

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Let S be the subspace of P3 consisting of all polynomials p(x) such that p(0) = 0 and let T be the subspace of all polynomials q(x) such that q(1) = 0.

a) The two vectors are linearly independent and span S which means {x, \(x^{2}\)} forms a basis for S.

b) The two vectors are linearly independent and span T which means \({(x -1),(x - 1)^2}\)forms a basis for T.

c) The vector is linearly independent and spans S∩T which means {x(x−1)} forms a basis for S ∩ T.

We have the information from the question:

Let S be the subspace of \(P_3\) consisting of all polynomials p(x).

We have:

p(0) = 0 and let T be the subspace of all polynomials q(x) such that q(1) = 0.

a) S is all polynomials of the form p(x) = \(ax^2 + bx\) where a, b are

real numbers.

p(0) = \(a(0)^2 + b(0)\) = 0 for all a, b.

I propose that {x, \(x^{2}\)} forms a basis for S.

We must show that:

The vectors x and \(x^{2}\) are linearly independent and span S.

To show they are linearly independent we must show that:

\(\alpha _1(x^2) + \alpha _2(x) = 0(x^2) + 0(x)\)

Only has the solution :

\(\alpha _1=\alpha _2=0\)

Upon grouping the terms we find:

\(\alpha _1=0\\\\\alpha _2=0\)

Thus the two vectors are clearly linearly independent.

Now to show that the two vectors span S we must show that any element

in S which I will represent by p(x) = ax^2 + bx can be written as:

\(\alpha _1(x^2) + \alpha _2(x) = ax^2 + bx\)

where, \(\alpha _1,\alpha _2\) are scalar  vectors.

Upon grouping the terms we find that:

\(\alpha _1=a\\\\\alpha _2=b\)

With this solution we have:

\(\alpha _1(x^2) + \alpha _2(x) = ax^2 + bx\)

which means the two vectors span S.

Thus, the two vectors are linearly independent and span S which means {x, \(x^{2}\)} forms a basis for S.

b)T is all polynomials of the form :

\(q(x) = a(x - 1)(bx + c) =abx^2 + acx - abx - ca = ab(x^2) + (ac - ab)x - ac\)where a, b, c are real numbers.

This is because q(1) = a(1 − 1)(b + c) = 0 for all a, b, c.

Let s = ab and t = ac.

Now we have that T is all polynomials of the form

\(q(x) = sx^2 + (t - s)x - t\)

\({(x - 1),(x - 1)^2}\)forms a basis for S.

In order to confirm this we must show that the vectors x − 1 and \((x - 1)^2\)are linearly independent and span S.

To show they are linearly independent we must show that:

\(\alpha _1((x -1)^2) + \alpha _2(x - 1) = 0(x - 1)(0(x) + 0)\)

only has the solution α1 = α2 = 0

Upon grouping the terms we find:

\(\alpha _1=0\\\\\alpha _2=0\)

Thus the two vectors are clearly linearly independent.

Now to show that the two vectors span T we must show that any element

in T which I will represent by \(q(x) = sx^2 + (t - s)x - t\) can be written as:

\(\alpha _1((x - 1)^2) + \alpha _2(x - 1) = sx^2 + (t - s)x - t\)

Where, \(\alpha _1,\alpha _2\) are scalars.

Upon grouping the terms we find that:

\(\alpha _1=s\\\\\alpha _2=s+t\)

With this solution we have:

\(sx^2 + (t - s)x - t = sx^2 + (t - s)x - t\)

which means the two vectors span T

Thus, the two vectors are linearly independent and span T which means \({(x -1),(x - 1)^2}\)forms a basis for T.

c)  S∩T is all polynomials of the form \(c(x) = a(x-1)(bx) = abx^2-abx\)

where a, b are real numbers.

This is because \(c(0) = a(0 - 1)^2\)

(b(0)) = 0 and

c(1) =\(a(1 - 1)^2\)

(b(1)) = 0 for all a, b.

Let ab = t

This means S∩T is all polynomials of the form \(c(x) = tx^2-tx = tx(x-1).\)

I propose that {x(x − 1)} forms a basis for S ∩ T.

Now, we must show that the vector x(x − 1) is linearly independent and spans S ∩ T.

To show it is linearly independent we must show that:

\(\alpha _1\)(x(x − 1)) = 0(x(x − 1))

only has the solution \(\alpha _1\) = 0.

Upon grouping the terms we find:

\(\alpha _1\) = 0

Thus the two vectors are clearly linearly independent.

Now to show that the vector spans S ∩ T we must show that any element

in S ∩ T which I will represent by c(x) = tx(x − 1) can be written as:

\(\alpha _1\)(x(x − 1)) = tx(x − 1).

where \(\alpha _1\) is a scalar.

Upon grouping the terms we find that:

\(\alpha _1\) = t

With this solution we have:

tx(x − 1) = tx(x − 1)

which means the vector spans S ∩ T.

Thus, the vector is linearly independent and spans S∩T which means {x(x−1)} forms a basis for S ∩ T.

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The centre line of a control chart is calculated by computing the average (mean) of the data for every period.

In control chart analysis, the centre line represents the central tendency or average value of the process being monitored. It is typically obtained by calculating the mean of the data points collected over a specific period. The purpose of the centre line is to provide a reference point against which the process performance can be compared. Any data points falling within acceptable limits around the centre line indicate that the process is stable and under control.

The calculation of the centre line involves summing up the values of the data points and dividing it by the number of data points. This average is then plotted on the control chart as the centre line. By monitoring subsequent data points and their distance from the centre line, deviations and trends in the process can be identified. Deviations beyond the control limits may indicate special causes of variation that require investigation and corrective action. Therefore, the centre line is a critical element in control chart analysis for understanding the baseline performance of a process and detecting any shifts or changes over time.

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The transfer price set by the selling division should be equal to the selling price less the variable costs plus the foregone contribution to the organization of making the transfer internally. Option D.

The transfer price refers to the price at which goods or services are transferred between divisions within the same organization. When the selling division sets the transfer price, it needs to consider various factors. Option D, selling price less the variable costs plus the foregone contribution to the organization of making the transfer internally, is the most appropriate choice.

The selling price less the variable costs ensures that the selling division covers its direct costs associated with the transferred goods or services. This ensures that the division remains financially viable and does not incur losses. However, it is also important to consider the opportunity cost of making the transfer internally.

The foregone contribution to the organization represents the potential profit or contribution that the selling division could have made if it had sold the goods or services to external customers instead of transferring them internally. By including this foregone contribution in the transfer price, the selling division accounts for the potential value it could have added to the organization.

In conclusion, the transfer price set by the selling division should consider both the variable costs associated with the transfer and the foregone contribution to the organization. Option D, selling price less the variable costs plus the foregone contribution to the organization of making the transfer internally, captures both of these factors and provides a comprehensive approach to setting the transfer price.

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A truck can carry either 576 1-foot boxes, 72 2-foot boxes, or 21 3-foot boxes.

The combination of boxes that will result in the highest payment for one truckload is 89 1-foot boxes, 3 2-foot boxes, and 3 3-foot boxes, for a total payment of $3,422.

To find how many boxes of one type will fill a truck, calculate the volume of the truck and divide it by the volume of one box.

Volume of the truck = 12 ft × 6 ft × 8 ft = 576 cubic feet

Volume of a 1-foot box = 1 ft × 1 ft × 1 ft = 1 cubic foot

Number of 1-foot boxes that will fill the truck = 576 cubic feet / 1 cubic foot = 576 boxes

Volume of a 2-foot box = 2 ft × 2 ft × 2 ft = 8 cubic feet

Number of 2-foot boxes that will fill the truck = 576 cubic feet / 8 cubic feet = 72 boxes

Volume of a 3-foot box = 3 ft × 3 ft × 3 ft = 27 cubic feet

Number of 3-foot boxes that will fill the truck = 576 cubic feet / 27 cubic feet = 21.33 boxes (rounded down to 21 boxes)

Therefore, a truck can carry either 576 1-foot boxes, 72 2-foot boxes, or 21 3-foot boxes.

To determine the combination of box types that will result in the highest payment for one truckload, calculate the total payment for each combination of box types.

Let x be the number of 1-foot boxes, y be the number of 2-foot boxes, and z be the number of 3-foot boxes in one truckload.

The volume of the boxes in one truckload is:

V = x(1 ft)³ + y(2 ft)³ + z(3 ft)³

V = x + 8y + 27z

The payment for one truckload is:

P = 5x + 25y + 100z

To maximize P subject to the constraint that the volume of the boxes does not exceed the volume of the truck:

x + 8y + 27z ≤ 576

Use the method of Lagrange multipliers to solve this optimization problem:

L(x, y, z, λ) = P - λ(V - 576)

L(x, y, z, λ) = 5x + 25y + 100z - λ(x + 8y + 27z - 576)

Taking partial derivatives and setting them equal to zero:

∂L/∂x = 5 - λ = 0

∂L/∂y = 25 - 8λ = 0

∂L/∂z = 100 - 27λ = 0

∂L/∂λ = x + 8y + 27z - 576 = 0

From the first equation, we get λ = 5.

Substituting into the second and third equations, y = 25/8 and z = 100/27. Since x + 8y + 27z = 576, x = 268/3.

Round these values to the nearest integer because no fraction for a box. Rounding down, x = 89, y = 3, and z = 3.

Therefore, the combination of boxes that will result in the highest payment for one truckload is 89 1-foot boxes, 3 2-foot boxes, and 3 3-foot boxes, for a total payment of $3,422.

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Answer:

50

Step-by-step explanation:

15 is 30% of 50

According to the statement the total probability is approximately 0.026, which is the second option in the list provided.

The probability that Brandon will earn a free homework pass can be found using the binomial probability formula. In this case, the number of trials (n) is 10, the probability of success (p) is 1/5 (since there are five answer choices per problem), and the probability of failure (q) is 4/5. We need to calculate the probability of getting at least 5 correct answers, meaning we will consider 5, 6, 7, 8, 9, and 10 correct answers.
The binomial probability formula is:
P(x) = C(n, x) * (p^x) * (q^(n-x))
Where P(x) is the probability of x successes, C(n, x) is the number of combinations of n items taken x at a time, and p and q are the probabilities of success and failure, respectively.
Calculating the probabilities for each of the desired outcomes (5 to 10 correct answers) and summing them up will give us the probability of Brandon earning a free homework pass. After doing the calculations, the total probability is approximately 0.026, which is the second option in the list provided.

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Answer:

0 solutions

Step-by-step explanation:

when the 2 equations aresolved they get cancelled off each other.

Answer:

Infinite solutions

Step-by-step explanation:

got it right

There is only one different square root of A.(a) To find the square roots of matrix A = [2 2; 2 2]: Let's consider two matrices B1 and B2: B1 = [1 1; 1 1]; B2 = [-1 -1; -1 -1].

Now, let's check if BB = A for each matrix: B1B1 = [1 1; 1 1] * [1 1; 1 1] = [2 2; 2 2] = A; B2B2 = [-1 -1; -1 -1] * [-1 -1; -1 -1] = [2 2; 2 2] = A. Both B1 and B2 satisfy BB = A, so they are two possible square roots of matrix A. (b) For matrix A = [5 0; 0 9], let's consider a matrix B: B = [√5 0; 0 √9] = [√5 0; 0 3] .We can see that BB = [√5 0; 0 3] * [√5 0; 0 3] = [5 0; 0 9] = A.

Thus, there is only one different square root of A. (c) Not every 2x2 matrix has a square root. For a square root of a matrix to exist, the matrix must be positive definite or positive semidefinite. If the eigenvalues of the matrix are negative or complex, there won't be any real square root. Additionally, if the matrix has zero eigenvalues with multiplicities greater than one, it may not have a unique square root. Therefore, the existence of a square root for a 2x2 matrix depends on its eigenvalues and their properties.

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Answer:

\(m = 2017\)  and \(m = -1000\)

Step-by-step explanation:

Given equations are (I am hoping it is 1017 as written by you)

From the first equation we get

\(mx-1000=1017 \\\\\rightarrow mx = 1017 + 1000\\\\\rightarrow mx = 2017\\\\\rightarrow m = 2017/x\dots [1]\)

From the second equation we get
\(1017=m-1000x\\\\m-1000x = 1017\\\\m = 1000x + 1017 \dots[2]\)

Equating [1] and [2] we get

\(\dfrac{2017}{x} = 1000x + 1017\\\\\)

Multiply above equation throughout by x to get

\(2017 = 1000x^2+ 1017x\\\)

Subtract 2017 from both sides:
0 = 1000x^2 + 1017x - 2017\\\\

Switching sides:

\(1000x^2 + 1017x - 2017 = 0\\\\\)

This is a quadratic equation in x which can be solved by the quadratic formula, completing the square or factorization

Let's choosing factoring to solve
\(1000x^2 + 1017x - 2017 = 0\) can be factored as

\(\left(1000x^2-1000x\right)+\left(2017x-2017\right) = 0\)\\\\

Factor out 1000x from the first term and 2017 from the second term:

\(\rightarrow 1000x(x - 1) + 2017(x -1) = 0\)\\\\

Factor out common term x - 1:
\(\left(x-1\right)\left(1000x+2017\right)\\\\\)

This means either \(x - 1 = 0 \;or\; 1000x + 2017 = 0\)

giving two possible solutions

\(x - 1 = 0 \rightarrow \boxed{x = 1}\)

and

\(1000x + 2017 = 0 \rightarrow 1000x = - 2017 \rightarrow \boxed{ x = -\dfrac{2017}{1000}}\)

Use these two values of x in equation 1 to solve for possible values of m

At x = 1
\(m = \dfrac{2017}{1} = 2017\)

At
\(x = -\dfrac{2017}{1000}\)

\(m = \dfrac{2017}{-\dfrac{2017}{1000}}\)

When dividing by a fraction, just multiply the numerator by the reciprocal of the denominator
\(\dfrac{a}{\dfrac{b}{c}}=\dfrac{a\cdot \:c}{b}\)

\(m =\dfrac{2017}{-\dfrac{2017}{1000}}\\\\\\= -\dfrac{2017\cdot \:1000}{2017}\\\\\\= - 1000\)

So the possible values of m are
\(\text{m = 2017 \;and\; m = -1000}\)

Answer:

Step-by-step explanation:

2 hours will pass before they meet.

"Information available from the question"

In the question:

Aaron starts riding a bike at a rate of 3 mi/h on a road toward a store 20 mi away.

Zhang leaves the store when Aaron starts riding his bike, and he rides his bike toward Aaron along the same road at 2 mi/h.

Now, According to the question:

Total distance = 20mi

Aaron starts riding a bike at a rate of  = 3mi/h

Zhang rides his bike toward Aaron along the same road at  = 2mi/h

Let y be the number of hours for them to meet.

The expression is given as

3y + 2y = 20

solving for y, we have

5y = 20

y = 20/5

y = 4hrs

Therefore, 2 hours will pass before they meet.

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Meiyin saved 2/7 of her salary if Meiyin gave 2/7 of her salary to her father. She gave 3/5 of the remaining salary to her mother and saved the rest.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

Let's say Meiyin's salary is represented by the whole unit.

Meiyin gave 2/7 of her salary to her father, which means she kept 5/7 of her salary.

Out of the remaining 5/7, Meiyin gave 3/5 to her mother. So the fraction of her salary she gave to her mother is:

(5/7) * (3/5) = 3/7

Therefore, the fraction of her salary that she saved is:

1 - (2/7 + 3/7) = 1 - 5/7 = 2/7

Therefore,  Meiyin saved 2/7 of her salary.

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Step-by-step explanation:

area of rectangle= 9×4=36m^2

9-3=6 which is the base of the triangle

6×3=18

18÷2=9m^2

36+9=45m^2

answer: 45m^2

1a) Complementary solution: y(t) = c₁ + c₂e^t + c₃e^(-t) + c₄cos(t) + c₅sin(t)

1b) Particular solutions:

(i) yp(t) = (1/2)e^t

(ii) yp(t) = (1/5)t^2 + (2/15)

(iii) yp(t) = [(3/20)t^2 + (1/4)t + C₁]cos(2t) - [(3/40)t^3 + (1/8)t^2 + C₂]sin(2t)

2) Linear differential equation: y''(x) + 4y'(x) + 5y(x) = 0

1a) To find the complementary solution of y(5) - y(3) = f(t), we need to find the homogeneous solution of the differential equation y(5) - y(3) = 0. We assume that y(t) = e^(rt) and substitute it into the differential equation to obtain the characteristic equation:

r^5 - r^3 = 0

r = 0, ±1

Since there are three distinct roots, the complementary solution is of the form:

y(t) = c₁ + c₂e^t + c₃e^(-t) + c₄cos(t) + c₅sin(t)

where c₁, c₂, c₃, c₄, and c₅ are constants.

1b) Using the method of undetermined coefficients, we can determine the particular solution of y(5) - y(3) = f(t) for each of the given functions f(t):

(i) f(t) = e^t

Assuming yp(t) = Ae^t, we obtain:

A(e^(5t) - e^(3t)) = e^t

A = 1/2

yp(t) = (1/2)e^t

(ii) f(t) = 2t^2

2A(t^5 - t^3) + 6B(t^3 - t) + 6C(t^2 - 1) = 2t^2

A = 1/5, B = 0, C = 2/15

yp(t) = (1/5)t^2 + (2/15)

(iii) f(t) = (3t+2)cos(2t)

A''(t)cos(2t) + B''(t)sin(2t) + 4A'(t)sin(2t) - 4B'(t)cos(2t) = (3t+2)cos(2t)

A(t) = (3/20)t^2 + (1/4)t + C₁

B(t) = -(3/40)t^3 + (1/8)t^2 + C₂

yp(t) = [(3/20)t^2 + (1/4)t + C₁]cos(2t) - [(3/40)t^3 + (1/8)t^2 + C₂]sin(2t)

2) The given general solution is:

y(x) = C₁e^x + C₂xe^x + C₃e^(-x)cos(2x) + C₄e^(-x)sin(2x)

y'(x) = C₁e^x + C₂(e^x + xe^x) - C₃e^(-x)sin(2x) + C₄e^(-x)cos(2x)

y''(x) = C₁e^x + C₂(2e^x + xe^x) + C₃e^(-x)cos(2x) - C₄e^(-x)sin(2x)

(C₁ + C₂ + C₃ + C₄)e^x + (2C₂ + C₂x - C₃sin(2x) + C₄cos(2x))e^x + (-C₁ + 2C₂ - C₃cos(2x) - C₄sin(2x))e^(-x) = 0

Since this equation must hold for all values of x, we obtain the following system of equations:

C₁ + C₂ + C₃ + C₄ = 0

2C₂ - C₁ - C₃cos(2x) - C₄sin(2x) = 0

C₂x - C

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