Let A, B, and C be arbitrary sets within a universal set, U. For each of the following statements, either prove that the statement is always true or show a counterexample to prove it is not always true. When giving a counterexample, you should define the three sets explicitly and say what the left-hand and right-hand sides of the equation are for those sets, to make it clear that they are not equal.

(A \ B) × C = (A × C) \ (B × C)

I suspect there's a typo in the question, because [tex]y_1=x[/tex] is *not* a solution to the corresponding homogeneous equation. We have [tex]{y_1}'=1[/tex] and [tex]{y_1}''=0[/tex], so the ODE reduces to

[tex]0+7x+5x=12x\neq0[/tex]

Let [tex]y=x^m[/tex], then [tex]y'=mx^{m-1}[/tex] and [tex]y''=m(m-1)x^{m-2}[/tex], and substituting these into the (homogeneous) ODE gives

[tex]m(m-1)x^m+7mx^m+5x^m=0\implies m(m-1)+7m+5=m^2+6m+5=(m+5)(m+1)=0[/tex]

which then admits the characteristic solutions [tex]y_1=\dfrac1x[/tex] and [tex]y_2=\dfrac1{x^5}[/tex].

Now to find a solution to the non-homogeneous ODE. We look for a solution of the form [tex]y(x)=v(x)y_1(x)[/tex] or [tex]y(x)=v(x)y_2(x)[/tex].

It doesn't matter which one we start with, so let's use the first case. We get derivatives [tex]y'=x^{-1}v'-x^{-2}v[/tex] and [tex]y''=x^{-1}v''-2x^{-2}v'+2x^{-3}v[/tex]. Substituting into the ODE yields

[tex]x^2(x^{-1}v''-2x^{-2}v'+2x^{-3}v)+7x(x^{-1}v'-x^{-2}v)+5x^{-1}v=x[/tex]

[tex]xv''+5v'=x[/tex]

Substitute [tex]w=v'[/tex], so that [tex]w'=v''[/tex] and

[tex]xw'+5w=x[/tex]

which is linear in [tex]w[/tex], and we can condense the left side as the derivative of a product after multiplying both sides by [tex]x^4[/tex]:

[tex]x^5w'+5x^4=x^5\implies(x^5w)'=x^5\implies x^5w=\dfrac{x^6}6+C\implies w=\dfrac x6+\dfrac C{x^5}[/tex]

Integrate to solve for [tex]v[/tex]:

[tex]v=\dfrac{x^2}{12}+\dfrac{C_1}{x^4}+C_2[/tex]

Then multiply both sides by [tex]y_1=\dfrac1x[/tex] to solve for [tex]y[/tex]:

[tex]y=\dfrac x{12}+\dfrac{C_1}{x^5}+\dfrac{C_2}x[/tex]

so we found another fundamental solution [tex]y_3=x[/tex] that satisifes this ODE.

Answer:

6 people

Step-by-step explanation:

Suppose A represents the event of going on Saturday,

B represents the event of going on Sunday,

According to the question,

n(A)=11

n(B)=14

n(A∪B)=20

We know that,

n(A∪B) = n(A) + n(B) - n(A∩B)

By substituting values,

20 = 11 + 14 - n(A∩B)

⇒ n(A∩B) = 25 - 20 = 5,

Hence, the number of people who can only go to the game on Saturday = n(A) - n(A∩B) = 11 - 5 = 6.

Audrey can visit the six cities in which she plans to conduct book signings and return to her starting point in 720 different ways. This is because of the mathematical principle of permutations.

Audrey's problem deals with permutations because the order of the places she visits matters. In general, the number of ways to arrange 'n' items (in Audrey's case, 'n' cities) in a specific order is given by 'n things taken n at a time' which is mathematically represented as n! (n factorial). In this case, Audrey is visiting 6 cities (Knoxville, Chattanooga, Chapel Hill, Charlotte, Raleigh, and Richmond), and then returning to her original city, Wilmington. So, the number of ways she can visit these cities can be represented as 6!, which equals 720.

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Answer:

Step-by-step explanation:

Correlation means that two or more events happen together. They are related to one another by being caused by the same thing.

Causation has a definite order. The first event has some cause that is comes before the second event. One event caused the other.

Answer:

Step-by-step explanation:

1. Greater density means the sphere has more mass in the same volume. The volume of water that must be displaced to equal that increased mass must be increased, causing the water level to rise.

__

2. Less mass means less water must be displaced to equal the mass of the new sphere, causing the water level to fall.

__

3. More mass is the same as higher density (see 1). The water level will rise.

Archimedes' principle states that the upward force acting on a body floating or immersed in a fluid is equal to the weight of the displaced fluid

The level of the water in the three situations are as follows;

Situation 1; Falls or stays the same, E: f or s

Situation 2; Falls, B: f

Situation 3, Rises A: r

The reason for the above selection is as follows;

The given details of the arrangements are;

The mass of the solid homogeneous sphere = m₀

The radius of the sphere = r₀

The density of the sphere = ρ₀

The location the sphere is placed = Floating in a container of water

The required parameter;

The provision of an estimate of the water level when the sphere is replaced with a new sphere with different physical parameters

Notation;

r = The water level rises

f = The water level falls

s = The water level stays the same

Situation 1; The mass of the new sphere, m = m₀

The density of the new sphere, ρ > ρ₀

Here, the denser sphere of equal mass = Smaller sphere, r < r₀

if the sphere floats, then the volume of the water displaced is equal to the

mass of the sphere, which is therefore, equal to the volume of the water

displaced by the original sphere

Therefore, the water level remains the same, s

However, if the sphere sinks, then the water displaced is less than the

mass m = m₀, of the sphere and therefore, the level falls, f

Therefore, the correct option is E: f or s

Situation 2: The mass of the new sphere, m < m₀

The radius of the new sphere, r = r₀

Here, we have equal radius and therefore equal volume and lesser density

Given that the volume of the water displaced for a floating body is equal to

the weight of body, and that the mass of the new sphere is less than the

mass of the original sphere, the mass of the water displaced and therefore,

the volume of water displaced is less and therefore, the water level falls

The correct option is therefore B: f falls

Situation 3: The mass of the new sphere, m > m₀, and the radius r = r₀

therefore the new sphere is denser than the original sphere and the

therefore, the mass of the water displaced where the sphere floats is m >

m₀, which is more than the water displaced for the original sphere and the

level of water rises, r, and the correct option is A: r

Therefore;

In situation 1, we have option E: f or s

In situation 2, the correct option is B: f

In situation 3, the correct option is A: r

Learn more about Archimedes' principle here:

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Answer:

The flow rate is 267ml/hour

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Equations.

To solve this we first need to find out how many ml the Rottweiler needs over 12 hours. We do this by using the Rule of Three property.

[tex]\frac{40ml}{1kg} = \frac{x}{80kg}[/tex]

[tex]\frac{40ml*80kg}{1kg} =x[/tex]

[tex]3200ml = x[/tex]

So the Rottweiler needs 3200 ml over a 12 hour period. We now need to find the flow rate per hour. We can solve this by simply dividing 3200 ml by 12 hours.

[tex]3200ml / 12hours = 266.67ml/hour[/tex]

So the flow rate is 267 ml/hour (rounded to the nearest whole number)

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Answer:

a) Arthur Penn

Step-by-step explanation:

There are three feature films based on Bonnie and Clyde they are the following:

"The Bonnie Parker Story"  released in 1958 was directed by William Witney.

"Bonnie and Clyde" released in 1967 was directed by Arthur Penn.

Warren Beaty is primarily an actor who has directed six films including a tv movie and five feature films.

"The Highwaymen" was directed by John Lee Hancock released 2019.

Answer:

A. Arthur Penn

Step-by-step explanation:

Bonnie and Clyde A defining film of the New Hollywood generation was Bonnie and Clyde (1967). Produced by and starring Warren Beatty and directed by Arthur Penn, its combination of graphic violence and humor, as well as its theme of glamorous disaffected youth, was a hit with audiences.

To find the area of a triangle with given vertices, calculate the cross product of two vectors representing the sides of the triangle. The magnitude of this cross product gives the area of the parallelogram, and half of this value is the triangle's area.

The area of a triangle with vertices (1, 0, 0), (0, 2, 0), and (0, 0, 1) can be calculated using the cross product of two vectors that represent two sides of the triangle. First, we find the vectors AB and AC by subtracting the coordinates of the points:

Next, we calculate the cross product AB x AC:

|i    j    k|
|-1  2  0|
|-1  0  1|

This results in a new vector (2, -1, -1). The magnitude of this vector gives us the area of the parallelogram formed by vectors AB and AC.

Area of parallelogram = |(2, -1, -1)| = √(2^2 + (-1)^2 + (-1)^2) = √(6)

Since the area of the triangle is half the area of the parallelogram, we get:

Area of triangle = ½ √(6) = √(1.5).

Answer: 208

Step-by-step explanation:

Given : An advertising company wishes to estimate the population mean of the distribution of hours of television watched per household per day.

Standard deviation : [tex]2.8\text{ hours}[/tex]

Margin of error : [tex]\pm0.5\text{ hour}[/tex]

Significance level : [tex]\alpha=1-0.99=0.01[/tex]

Critical value : [tex]z_{\alpha/2}=2.576[/tex]

The formula to calculate the sample size is given by :-

[tex]n=(\dfrac{z_{\alpha/2}\sigma}{E})^2[/tex]

[tex]\Rightarrow\ n=(\dfrac{2.576\times2.8}{0.5})^2=208.09793536\approx208[/tex]

Hence, the minimum required sample size must be 208.

Answer:

last installment is $540

Step-by-step explanation:

principal amount (p) = $7500

rate (r) = 16.5 %

installment = $300

to find out

full payment is increased to pay off the loan and the last smaller payment is made one month after the last full payment

solution

we know monthly installment is $300 so amount will be paid i.e.

amount = $300×12×N ..............1

here N is no of installment

and we know amount formula i.e.

amount = principal ( 1+r/100)^N

put amount value and principal rate

300×12×N = 7500 ( 1+16.5/100)^N

(3600 ×N ) / 7500 = 1.165^N

0.48N = 1.165^N

by the graphical we will get N = 3.65

so 3.65 year

so as that put N in equation 1 we get

amount = $300×12× 3.65

amount = $13140

we can say there are 43 installment so remaining money is  $13140 - ($300 × 43 installment )

i.e. = $240 and last installment will be $300 + $240 = $540

so last installment is $540

ANSWER

[tex]36k^{8} -13{k}^{6} -64k^{4} + 30 {k}^{2} [/tex]

EXPLANATION

Recall the distributive property:

[tex](a + b + c)(d + e) = a(d + e) + b(d + e) + c(d + e)[/tex]

We apply this property multiple times to simplify

[tex](9k^{6}+8k^{4}-6k^{2})(4k^{2}-5)[/tex]

This implies that:

[tex]9k^{6}(4k^{2}-5)+8k^{4}(4k^{2}-5)-6k^{2}(4k^{2}-5)[/tex]

We apply the distributive property again:

This time: a(b+c)=ac+ab

[tex] \implies \: 9k^{6} \times 4k^{2}-5 \times 9 {k}^{6} +8k^{4} \times 4k^{2}-5 \times 8 {k}^{4} -6k^{2} \times 4k^{2} + 5 \times 6 {k}^{2} [/tex]

[tex]\implies \: 36k^{8} -45{k}^{6} +32k^{6} -40 {k}^{4} -24k^{4} + 30 {k}^{2} [/tex]

[tex]\implies 36k^{8} -13{k}^{6} -64k^{4} + 30 {k}^{2} [/tex]

NB: [tex]k^{n}\times{k}^{m}=k^{m+n} [/tex]

Answer:

The expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd

Step-by-step explanation:

Let's remember the logical operations with even and odd numbers

odd*odd=odd

even*even=even

odd*even=even

odd-odd=even

even-even=even

even-odd=odd

Now, the original expression is:

[tex]m^3-n^3[/tex] which can be expressed as:

[tex](m*(m*m))-(n*(n*n))[/tex]

If m and n are both odd, then:

[tex](m*(m*m))=odd*(odd*odd)=odd*(odd)=odd[/tex]

[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]

Then, [tex](m*(m*m))-(n*(n*n))=odd-odd=even[/tex]

If m and n are both even, then:

[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]

[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]

Then, [tex](m*(m*m))-(n*(n*n))=even-even=even[/tex]

Finally if one of them is even, for example m, and the other is odd, for example n, then:

[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]

[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]

Then, [tex](m*(m*m))-(n*(n*n))=even-odd=odd[/tex]

In conclusion, the expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd. If one of them is even and the other one is odd, then the expression is odd.

Answer:

C

Step-by-step explanation:

A. First two rectangles are dilations because

[tex]\dfrac{2}{4}=\dfrac{4}{8}=0.5[/tex]

B. Second two rectangles are dilations because

[tex]\dfrac{2}{4}=\dfrac{3}{6}=0.5[/tex]

C. Third two rectangles are not dilations because

[tex]\dfrac{3}{4}\neq \dfrac{2}{3}[/tex]

D. Fourth two rectangles are dilations because

[tex]\dfrac{3}{4}=\dfrac{1.5}{2}=0.75[/tex]

Answer:

c please correct me if im wrong

Step-by-step explanation:

Answer:

The number of dosage is 0.13.

Step-by-step explanation:

Here, the given function that represents the blood pressure,

[tex]B(x)=0.06x^2 - 0.3x^3[/tex]

Where, x is the number of dosage in cubic centimeters​,

Differentiating the above function with respect to x,

[tex]B'(x)=0.12x-0.9x^2[/tex]

For maximum or minimum blood pressure,

[tex]B'(x)=0[/tex]

[tex]0.12x-0.9x^2=0[/tex]

[tex]-0.9x^2=-0.12x[/tex]

[tex]x=\frac{0.12}{0.9}=\frac{2}{15}[/tex]

Again differentiating B'(x) with respect to x,

[tex]B''(x)=0.12-1.8x[/tex]

Since, at x = 2/15,

[tex]B''(\frac{2}{15})=0.12-1.8(\frac{2}{15})=0.12-0.24=-0.12=\text{Negative value}[/tex]

So, at x = 2/15 the value of B(x) is maximum,

Hence, the number of dosage at which the resulting blood pressure is maximized = 2/15 = 0.133333333333 ≈ 0.13

The maximum blood pressure results from a dosage of approximately 0.13 cubic centimeters, based on the mathematical model given in the problem.

To find the maximum blood pressure using the formula B (x) = 0.06 x^2 - 0.3 x^3, we need to first find the derivative of this equation, as the maximum point on any curve happens when its derivative equals zero.

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Answer:

z=2.27

Step-by-step explanation:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

where z is the deviation from mean.

mean (μ) = 35 inches

standard deviation (σ) = 11 inches

last year snow fall (x) = 60 inches

[tex]z=\frac{x-\mu}{\sigma}[/tex]

[tex]z=\frac{60-35}{11}[/tex]

z=2.27

now, the standard deviation for the 60 inches snow from the mean is calculated to be 2.27

Answer:

6

Step-by-step explanation:

96x5

Answer:

32x(2)         (squared)

Step-by-step explanation:

GCF of 96 and 64:

  64 = (2)(2)(2)(2)(2)(2)

  96 = (2)(2)(2)(2)(2)(3)

  GCF = (2)(2)(2)(2)(2) = 32

GCF of x5 and x2:

x5 = (x)(x)(x)(x)(x)

x2 = (x)(x)

GCF = (x)(x) = x2

The answer is:

Center: (1,-2)

Radius: 4 units.

To solve the problem, using the given formula of a circle, we need to find its standard equation form which is equal to:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]

Where,

"h" and "k"are the coordinates of the center of the circle and "r" is its radius.

So, we need to complete the square for both variable "x" and "y".

The given equation is:

[tex]x^2+y^2-2x+4y-11=0[/tex]

So, solving we have:

[tex]x^2+y^2-2x+4y=11[/tex]

[tex](x^2-2x+(\frac{2}{2})^{2} )+(y^2+4y+(\frac{4}{2})^{2})=11+(\frac{2}{2})^{2} +(\frac{4}{2})^{2}\\\\(x^2-2x+1)+(y^2+4y+4)=11+1+4\\\\(x^2-1)+(y^2+2)=16[/tex]

[tex](x^2-1)+(y^2-(-2))=16[/tex]

Now, we have that:

[tex]h=1\\k=-2\\r=\sqrt{16}=4[/tex]

So,

Center: (1,-2)

Radius: 4 units.

Have a nice day!

Note: I have attached a picture for better understanding.

To calculate the value of 715 × 211, you can use the standard multiplication method by multiplying each digit of the two numbers and summing up the results.

To find the value of 715 × 211, you can use the standard multiplication method. Start by multiplying the ones digit of 715 (5) by each digit of 211 (1, 1, and 2), and write down the results. Then, multiply the tens digit of 715 (1) by each digit of 211, and write down the results one place to the left of the previous results. Finally, multiply the hundreds digit of 715 (7) by each digit of 211 and write down the results two places to the left. Sum up the columns and you will get the final product.

Here's how it looks:

   715
× 211
--------
 715
 1430  
+1425
--------
 150665

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Answer:

  The junction law is based on the conservation of charge.

Step-by-step explanation:

Kirchhoff's current law, or junction law, (1st Law) states that current flowing into a node (or a junction) must be equal to current flowing out of it. This is a consequence of charge conservation—charge is not created or destroyed in a closed system.

I hope I've helped

In this photo you can find de probabilities

Step-by-step explanation:

There are 20 chips in total.

P(red) = 8/20 = 2/5

P(not green) = 16/20 = 4/5

P(yellow or green) = 9/20

Answer:

The equation 2.5x -10.5 = 64(0.5^x) is true when x=5

Step-by-step explanation:

we need to solve the equation 2.5x -10.5 = 64(0.5^x)

We have to put the given values of x in the functions f(x) and g(x) and find their values

x     f(x) = 2.5x -10.5            g(x) = 64(0.5^x)

2     2.5(2)-10.5 = -5.5          64(0.5^2) = 16      

3     2.5(3) - 10.5 = -3           64(0.5^3) = 8

4     2.5(4) - 10.5 = -0.5        64(0.5^4) = 4

5     2.5(5) - 10.5 = 2            64(0.5^5) = 2

6     2.5(6) - 10.5 = 4.5         64(0.5^6) = 1

So, we need to solve the equation 2.5x -10.5 = 64(0.5^x)

This holds when x = 5 as shown in the table above.

Answer:

  yes

Step-by-step explanation:

The line intersects each parabola in one point, so is tangent to both.

__

For the first parabola, the point of intersection is ...

  y^2 = 4(-y-1)

  y^2 +4y +4 = 0

  (y+2)^2 = 0

  y = -2 . . . . . . . . one solution only

  x = -(-2)-1 = 1

The point of intersection is (1, -2).

__

For the second parabola, the equation is the same, but with x and y interchanged:

  x^2 = 4(-x-1)

  (x +2)^2 = 0

  x = -2, y = 1 . . . . . one point of intersection only

___

If the line is not parallel to the axis of symmetry, it is tangent if there is only one point of intersection. Here the line x+y+1=0 is tangent to both y^2=4x and x^2=4y.

_____

Another way to consider this is to look at the two parabolas as mirror images of each other across the line y=x. The given line is perpendicular to that line of reflection, so if it is tangent to one parabola, it is tangent to both.

Answer: 0.116

Step-by-step explanation:

The Poisson distribution probability formula is given by :-

[tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex], where \lambda is the mean of the distribution and x is the number of success

Given : The number of inclusions in one cubic millimeter = 3.2

Then , the number of inclusions in two cubic millimeters=[tex]\lambda=2\times3.2=6.4[/tex]

Now, the probability of exactly four inclusions in 2.0 cubic millimetres is given by :-

[tex]P(X=4)=\dfrac{e^{-6.4}(6.4)^4}{4!}\\\\=0.11615127195\approx0.116[/tex]

Hence, the probability of exactly four inclusions in 2.0 cubic millimetres = 0.116

Answer:

Dear, Have a look at pic

Answer:

Given relations defined on the set {a, b, c, d},

S= { (a, b), (a, c), (c, d), (c, a)}

R={ (b, c), (c, b), (a, d), (d, b)},

Since, SoR(x) = S(R(x)),

So, SoR(a) = S(R(a)) = S(d) = ∅,

SoR(b) = S(R(b)) = S(c) = d and a,

SoR(c) = S(R(c)) = S(b) = ∅,

SoR(d) = S(R(d)) = S(b) = ∅,

Thus, SoR = { (b,d), (b,a) }

RoS(a) = R(S(a)) = R(b) = c and RoS(a) = R(S(a)) = R(c) = b,

RoS(b) = R(S(b)) = R(∅) = ∅,

RoS(c) = R(S(c)) = R(d) = b and RoS(c) = R(S(c)) = R(a) = d

RoS(d) = R(S(d)) = R(∅) = ∅,

Thus, RoS = { (a, c), (a, b), (c,d), (c, b) },

SoS(a) = S(S(a)) = S(b) = ∅ and SoS(a) = S(S(a)) = S(c) = d and a

SoS(b) = S(S(b)) = S(∅) = ∅,

SoS(c) = S(S(c)) = S(d) = ∅ and SoS(c) = S(S(c)) = S(a) = b and c

SoS(d) = S(S(d)) = S(∅) = ∅,

SoS = { (a, d), (a, a), (c, b), (c, c) }

The composition of relations S and R mentioned in the question are SoR: { (a, c), (c, b)}, RoS: { (b, d), (a, b)} and SoS: { (a, d), (c, b)}.

The question is asking for the composition of relations. So, composition of relations S and R, denoted as 'SoR' or 'S ◦ R', is the set of ordered pairs where the first element is related to the second element through the combination of relations S and R. In this case the relations S and R on the set {a, b, c, d} are: S= { (a, b), (a, c), (c, d), (c, a)} and R={ (b, c), (c, b), (a, d), (d, b)}.

By the rule of composition SoR will be: { (a, c), (c, b)}.

Similarly, for RoS will be: { (b, d), (a, b)}.

And for SoS it will be: { (a, d), (c, b)}.

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Answer:  216

Step-by-step explanation:

Given : We like to make a salad that consists of​ lettuce, tomato,​ cucumber, and onions.

The number of varieties of lettuce = 9

The number of varieties of tomatoes = 4

The number of varieties of cucumbers = 2

The number of varieties of onions = 3

Now, the number of different salads we can make is given by :-

[tex]9\times4\times2\=216[/tex]

Hence, we can make 216 different types of salads.

Answer:

415

Step-by-step explanation:

Confidence Level = 98%

Z-value for this confidence level = z = 2.326

Margin of error = E = 0.08

Mean = u = 6.6

Standard deviation = [tex]\sigma=0.7[/tex]

Required Sample Size = n = ?

The formula for margin of error is:

[tex]E=z\frac{\sigma}{\sqrt{n}}[/tex]

Re-arranging the equation for n, and using the given values we get:

[tex]n=(\frac{z\sigma}{E} )^{2}\\\\ n=(\frac{2.326 \times 0.7}{0.08} )\\\\ n=415[/tex]

Thus, the minimum sample size required to create the specified confidence interval is 415

The minimum sample size required to construct a 98% confidence interval with an error of no more than 0.08 is 255.

To determine the minimum sample size required to construct a 98% confidence interval with an error of no more than 0.08, we can use the formula:

n = (Z * sigma / E) ^ 2

where n is the sample size, Z is the Z-score corresponding to the desired confidence level, sigma is the standard deviation, and E is the desired margin of error.

In this case, the Z-score for a 98% confidence level is approximately 2.33. Substituting the given values of sigma = 0.7 and E = 0.08 into the formula, we can calculate the minimum sample size:

n = (2.33 * 0.7 / 0.08) ^ 2

n ≈ 254.43

Rounding up to the next integer, the minimum sample size required is 255.

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Suppose [tex]y_p=a_0+a_1t[/tex] is a solution to the ODE. Then [tex]{y_p}'=a_1[/tex] and [tex]{y_p}''=0[/tex], and substituting these into the ODE gives

[tex]a_1-4(a_0+a_1t)=7t+5\implies\begin{cases}-4a_1=7\\-4a_0+a_1=5\end{cases}\implies a_0=-\dfrac{27}{16},a_1=-\dfrac74[/tex]

Then the particular solution to the ODE is

[tex]y_p=-\dfrac{27}{16}-\dfrac74t[/tex]