The general solution to this ODE: y"" – y' – 2y = ( – 5t^2 + 4t – 2)e^-3t = is: y(t) = ____ Use c1 and c2 for any undetermined constants in your solution.
Answers
Using c1 and c2 as undetermined constants, we express the general solution.
To solve the given second-order linear non-homogeneous differential equation:
\(y'' - y' - 2y = (-5t^2 + 4t - 2)e^{(-3t)}\),
we first find the complementary solution by solving the corresponding homogeneous equation:
y'' - y' - 2y = 0.
Assuming a solution of the form y(t) = e^(rt) and substituting it into the homogeneous equation, we obtain the characteristic equation:
\(r^2 - r - 2 = 0.\)
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring is the most convenient:
(r - 2)(r + 1) = 0.
Setting each factor to zero gives us the roots:
r - 2 = 0 => r = 2,
r + 1 = 0 => r = -1.
Therefore, the two roots of the characteristic equation are r1 = 2 and r2 = -1.
The complementary solution of the homogeneous equation is given by:
y_c(t) = \(c1e^{(2t)}+ c2e^{(-t)}\)
where c1 and c2 are undetermined constants.
Now, to find the particular solution of the non-homogeneous equation, we can use the method of undetermined coefficients. We assume a particular solution of the form:
y_p(t) = \((At^2 + Bt + C)e^{(-3t)}\),
where A, B, and C are constants to be determined.
Taking the derivatives of y_p(t), we have:
y'_p(t) = \((-3At^2 - (6A + B)t - 3B + C)e^{(-3t)}\),
y''_p(t) = (\(6At^2 + (18A + 6B)t + (9A - 6B - 3C))e^{(-3t)}\).
Substituting these derivatives and y_p(t) into the non-homogeneous equation, we get:
\((6At^2 + (18A + 6B)t + (9A - 6B - 3C))e^{(-3t)} - (-3At^2 - (6A + B)t - 3B + C)e^{(-3t)} - 2(At^2 + Bt + C)e^{(-3t)} = (-5t^2 + 4t - 2)e^{(-3t)}\).
Simplifying, we have:
\((6A + 3A - 2A)t^2 + (18A + 6B + 6A + B - 2B - 4A)t + (9A - 6B - 3C + 3B + C - 2C) = (-5t^2 + 4t - 2)\).
Matching the coefficients of like terms on both sides of the equation, we have the following system of equations:
6A + 3A - 2A = -5,
18A + 6B + 6A + B - 2B - 4A = 4,
9A - 6B - 3C + 3B + C - 2C = -2.
Simplifying the system of equations, we get:
7A = -5,
20A + 5B = 4,
9A - 3C - 3B - C = -2.
Solving this system of equations gives A = -5/7, B = 94/35, and C = 11/35.
Therefore, the particular solution is:
y_p(t) = \((-5/7)t^2 + (94/35)t + (11/35)e^{(-3t)}\).
The general solution of the non-homogeneous equation is the sum of the complementary and particular solutions:
y(t) = y_c(t) + y_p(t)
= \(c1e^{(2t)} + c2e^{(-t)} + (-5/7)t^2 + (94/35)t + (11/35)e^{(-3t)}\).
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Related Questions
the anova procedure is a statistical approach for determining whether the means of . a. more than two samples are equal b. two or more populations are equal c. two samples are equal d. two or more samples are equal
Answers
The means of two or more populations being equal is determined by a statistical approach for the ANOVA procedure. Option B is correct.
The ANOVA (Analysis of Variance) procedure is a statistical method used to determine whether there is a significant difference between the means of two or more groups. To statistically test the equality of means ANOVA uses F-tests.
The repeated-measures ANOVA is a two-stage process that is described as an analysis of dependencies. This test is used to prove an assumed cause-effect relationship between variables. The conditions that must be met for the results of an ANOVA are Independence, Random Sampling, Large Sample Size, and Normality.
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Write an equation of the line in slope-intercept form.
Ay
4
y =
(0, 2)
2
(3, 3)
4
AX
Answers
Equation of line in slope-intercept form is \(y = (\frac{1}{3})x + 2\).
Given in graph,
A(0,2) and B(3,3)
Slope-intercept form => y = mx + c
Slope, m = \(\frac{y2-y1}{x2-x1}\)
= \(\frac{3-2}{3-0}\)
= \(\frac{1}{3}\)
Therefore, y = (1/3)x + c
Put A(0,2) in above equation
2 = (1/3)*0 + c
2 = c
Hence, y = (1/3)x + 2.
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What is the value of x in the equation 4 x plus 8 y equals 40, when y equals 0.8?
4.6
8.4
10
12
Answers
4x + 6.4 = 40
4x = 33.6
x = 8.4
Find the area of the chape giving… pls help me
Answers
Answer:
Area = 8 sq cm
Step-by-step explanation:
This shape is a parallelogram. Its like a rectangle that got tipped over. Area is:
base × height.
We don't know the base that is on the bottom of the image as it is shown. But if you turn it sideways, the 4 can be the base and the 2 is the height.
Area = b × h
= 4 × 2
= 8
The area is 8 sq cm.
let f(x) = cx ln(cos(x)). for what value of c is f ′ 4 = 7?
Answers
The value of c: -0.0018.
How to find the value of c?We first find the derivative of f(x) using the product and chain rule:
f(x) = cx ln(cos(x))
f'(x) = c ln(cos(x)) + cx * (-sin(x)/cos(x))
f'(x) = c ln(cos(x)) - cx tan(x)
Then we can find the fourth derivative:
\(f''''(x) = c * (- 2sec^4(x) - 16sin^2(x)sec^2(x) + 12sin^4(x)sec^2(x)) - cx * (24sin^3(x)/cos^3(x) + 12sin(x)/cos(x))\)
Now we evaluate the fourth derivative at x = 4:
\(f''''(4) = c * (- 2sec^4(4) - 16sin^2(4)sec^2(4) + 12sin^4(4)sec^2(4)) - c*4 * (24sin^3(4)/cos^3(4) + 12sin(4)/cos(4))\)
We set this equal to 7 and solve for c:
\(7 = c * (- 2sec^4(4) - 16sin^2(4)sec^2(4) + 12sin^4(4)sec^2(4)) - c*4 * (24sin^3(4)/cos^3(4) + 12sin(4)/cos(4))\)
Using a calculator, we find that the value of c that satisfies this equation is approximately -0.0018.
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The angle measures of a triangle are shown in the diagram
what is the value of x?
pls hurry its due soon
Answers
Answer: Subtract the sum of the two angles from 180 degrees. The sum of all the angles of a triangle always equals 180 degrees. Write down the difference you found when subtracting the sum of the two angles from 180 degrees. This is the value of X.
Step-by-step explanation: To find the value of x, bring the variable to the left side and bring all the remaining values to the right side. Simplify the values to find the result.
Question 3 Find whether the vectorrs are parallel. (-2,1,-1) and (0,3,1)
a. Parallel
b. Collinearly parallel
c. Not parallel
d. Data insufficient
Answers
To determine whether the vectors (-2,1,-1) and (0,3,1) are parallel, we need to compare their direction. If they have different directions, they are not parallel. the correct answer is option c) Not parallel.
To check if two vectors are parallel, we can compare their direction vectors. The direction vector of a vector can be obtained by dividing each component of the vector by its magnitude. In this case, let's calculate the direction vectors of the given vectors.
The direction vector of (-2,1,-1) is obtained by dividing each component by the magnitude:
Direction vector of (-2,1,-1) = (-2/√6, 1/√6, -1/√6)
The direction vector of (0,3,1) is obtained by dividing each component by the magnitude:
Direction vector of (0,3,1) = (0, 3/√10, 1/√10)
Comparing the direction vectors, we can see that they are not equal. Therefore, the vectors (-2,1,-1) and (0,3,1) are not parallel. Hence, the correct answer is option c) Not parallel.
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On the first 4 of 5 test sections with possible scores between 0 and 20, Jocelyn scored 17, 20, 18, and 19. If she scored an odd number on the final portion of the test, what could have been her total grade
Answers
Jocelyn's total grade could be either 73 or 74, depending on the specific odd score she received on the final portion of the test.
We know that Jocelyn scored 17, 20, 18, and 19 on the first four sections of the test. Let's denote her score on the fifth and final section as x. Since Jocelyn scored an odd number on the final portion, x must be an odd number.
To calculate the total grade, we sum up the scores from all five sections:
Total grade = 17 + 20 + 18 + 19 + x
Given that x is an odd number, we can substitute some possible values to determine the potential total grades:
1. If x = 15:
Total grade = 17 + 20 + 18 + 19 + 15 = 89
2. If x = 17:
Total grade = 17 + 20 + 18 + 19 + 17 = 91
3. If x = 19:
Total grade = 17 + 20 + 18 + 19 + 19 = 93
Since we are looking for the possible total grades when Jocelyn scored an odd number on the final portion, the potential total grades could be 89, 91, or 93.
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Need step-by-step explanation. Will mark the brainliest. Basically explained this equation as if i am a 5 year old child. (with step of course)
Answers
attached is solution. please excuse handwriting.
break up summation into two smaller summation,
then use summation formula for E (x)
a lawn sprinkler waters a lawn in a circle as shown
Answers
Part A
I'm assuming you want the area of a circle with radius 17 yards.
We'll use the formula below
A = pi*r^2
A = (22/7)*17^2
A = (22*17^2)/7
A = 6358/7
A = 908 2/7
The sprinkler covers an area of about 908 2/7 square yards.
You'll type 908 in the left-most box
Then for the other two boxes, you'll have 2 up top and 7 down below to represent the fraction 2/7
The number 908 2/7 is a mixed number.
=======================================================
Part B
You mentioned that the radius reduces to 11 yards here. We'll use the same formula as above and pi = 22/7 approximately
A = pi*r^2
A = (22/7)*11^2
A = (22*11^2)/7
A = 2662/7
A = 380 2/7
The sprinklers can water an area of about 380 2/7 square yards.
This value is also a mixed number
380 is the value in the left most box (it's the whole part)
The other two boxes are filled out the same exact way as the previous part above.
Use the slope formula to find the slope of a line that passes through the points (-1,-8) and (-2, -10)
Answers
Let's take a look at the slope formula:
\(m=\frac{y_2-y_1}{x_2-x_1}\)Where
m is the slope
(x_1,y_1) is the first point
(x_2, y_2) is the second point
Points given are:
\(\begin{gathered} (x_1,y_1)=(-1,-8) \\ \text{and} \\ (x_2,y_2)=(-2,-10) \end{gathered}\)Let's substitute and find the slope. Shown below:
\(\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{-10-(-8)}{-2-(-1)_{}} \\ m=\frac{-10+8}{-2+1} \\ m=\frac{-2}{-1} \\ m=2 \end{gathered}\)THe slope is 2.
Victor spent $61 on some sandpaper for his model
cars. He bought 2 packages of the smallest-grain
sandpaper and spent the rest on the largest-grain
sandpaper. How many packages of the largest-
grain sandpaper did he buy?
Answers
Victor bought 10 packages of the largest-grain sandpaper.
What does "spent 50 balance 51" mean?Total money spent (20+15+9+6) = 50; total money left over (30+15+6+0) = 51. Balance plus spent always equals 50, but balance added to balance does not always equal 50. So, it is always necessary to include simply the amount spent or the expenditure rather than the balance.
Thus, we know how much he spent:
2S + (61 - 2S) = 61 - S
$1 on sandpaper with the biggest grit. We can condense this phrase as follows:
61 - S = 61 - 2S
Adding S to both sides, we get:
S = 0
Then we know that he spent:
2L + Ly = 61
Spending money on sandpaper. Additionally, since he purchased two packages of the finest sandpaper, the price of those two packages is:2S = 2L
We can substitute 2L for 2S in the first equation:
2L + Ly = 61
Simplifying, we get:
2L + L(2/3)L = 61
Multiplying both sides by 3/2, we get:
3L² = 91.5
Taking the square root of both sides, we get:
L ≈ 5.27
determine how many packages of the coarsest sandpaper Victor purchased:
2L + Ly = 61
2(5.27) + 5.27y = 61
10.54 + 5.27y = 61
5.27y = 50.46
y ≈ 9.56
Rounding to the nearest whole number, we get:
y = 10
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What is an expression for the sale price of a bracelet that has been discounted 35% from its
sticker price? Evaluate the expression for a sticker price of $60. Use the variable s for the sale
price and p for the sticker price.
Answers
The expression which represents the sales price of the bracelet in terms of the sticker price is; s = 0.65p and the evaluation for a sticker price of $60 is; $39.
Which expression represents the sales price of the bracelet in terms of the sticker price?It follows from the task content that the bracelet has been discounted 35% from its sticker price.
Hence, the sales price of the bracelet is therefore; (100 - 35)% of sticker price, p.
Therefore, we have;
s = (65/100) × p
s = 0.65p.
Ultimately, the evaluation for a sticker price of $60 is; s = 0.65(60).
s = $39.
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What is the growth factor when something is decreasing by:
15.7%
0.12%
Answers
When something decreases by 15.7%, the growth factor is about 1.182 and when something decreases by 0.12%, the growth factor is about 1.00012.
In math, what is the definition of multiplying?
Multiplication is a mathematical process that shows the amount of times a number has been added to itself. It is represented by the multiplication symbols (x) or (*). Division is a mathematical process that shows how many equal amounts add up to a given quantity.
If something decreases by 15.7%, the increase in the factor is 100 / (100 - 15.7) Equals 1.182.
As a result, when something decreases by 15.7%, the growth factor is about 1.182.
When something falls by 0.12%, the expansion factor is 100 / (100 - 0.12) = 1.00012.
As a result, when something decreases by 0.12%, the growth factor is about 1.00012.
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PLEASE HELP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
What type of polynomial is: −2/3b*3
A. quadratic
B. linear
C. quartic
D. cubic
Answers
The type of polynomial that - 2 / 3 x b³ is D. Cubic polynomial.
What is a cubic polynomial ?A cubic polynomial is a type of polynomial function in algebra that has a degree of three. Cubic polynomials can take many different forms and can have multiple real roots, complex roots, or no real roots at all.
A quadratic polynomial contains a degree of 2, a linear polynomial contains a degree of 1, and a quartic polynomial contains a degree of 4. In this case, the highest degree of the variable b is 3, which makes it a cubic polynomial.
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Simplify the product: (4x-3) (3x+1)
O A. x-2
OB. 12x²-3
O C. 12x-5x-3
OD. 12x³-9x² +4x-3
Answers
Answer:
\(\large\boxed{\mathtt{C) \ 12x^{2}-5x-3}}\)
Step-by-step explanation:
\(\textsf{For this problem, we are asked to multiply 2 Binomials.}\)
\(\textsf{We should use the FOIL Method.}\)
\(\large\underline{\textsf{FOIL Broken Down;}}\)
\(\textsf{F - Fronts}\)
\(\textsf{O - Outers}\)
\(\textsf{I - Inners}\)
\(\textsf{L - Last}\)
\(\large\underline{\textsf{For Example.}}\)
\(\mathtt{(2x+2y)(3x+3y)}\)
\(\mathtt{(F) \ 2x \times 3x = 6x^{2}}\)
\(\mathtt{(O) \ 2x \times 3y = 6xy}\)
\(\mathtt{(I) \ 2y \times 3x = 6xy}\)
\(\mathtt{(L) \ 2y \times 3y = 6y^{2}}\)
\(\textsf{Let's use the FOIL Method for our problem.}\)
\(\large\underline{\textsf{FOIL;}}\)
\(\mathtt{(4x-3)(3x+1)}\)
\(\mathtt{(F) \ 4x \times 3x = 12x^{2}}\)
\(\mathtt{(O) \ 4x \times 1 = 4x}\)
\(\mathtt{(I) \ -3 \times 3x = -9x}\)
\(\mathtt{(L) \ -3 \times 1 = -3}\)
\(\large\underline{\textsf{We should have;}}\)
\(\mathtt{12x^{2}+4x-9x-3}\)
\(\large\underline{\textsf{Combine Like Terms;}}\)
\(\large\boxed{\mathtt{C) \ 12x^{2}-5x-3}}\)
A diver dove from a board that was 6 3/4 feet above the water's surface, and descended until he was 8 1/3 feet below the water's surface. What was the total distance that the diver desceded?
Answers
Answer:
15 1/12 feet
Step-by-step explanation:
Well if he descended 8 1/3 feet below the water, that means he must've went from board to water which is 6 3/4 feet then down 8 1/3 feet
6 3/4 + 8 1/3
= 6 9/12 + 8 4/12
=15 1/12 feet
At its highest,the elevation of a county is 5,776 feet above sea level at its lowest point the elevation of the county is 9 feet below sea level
Answers
5,776-(-9)
Positive
5,785
Write an integer that represents the elevation of county feet below sea level.
Sixty-four reduced to a number is forty eight
now change it to a algebra numbers
Answers
The equation "Sixty-four reduced to a number is forty-eight" can be expressed algebraically or as algebra expression as: 64 - x = 48
In this equation, the variable "x" represents the unknown number that we are trying to find. By subtracting "x" from 64, we are reducing the value to 48.
Therefore, it can be translated into an algebraic equation:
64 - x = 48
Solving the equation allows us to determine the value of the variable "x," which represents the number we are looking for. The use of variables in algebraic expressions allows us to generalize problems and find solutions for different scenarios.
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28-3(2+4)+9
With explanation
Answers
Answer:
Go through PEMDAS
Step-by-step explanation:
28-3(2+4)+92+4=628-3(6)+96+9=1528-3+1528-3=2525+15=40Answer:
19
Step-by-step explanation:
Result:
28 - 3(2 + 4) + 9 = 19/1 = 19
Spelled result in words is nineteen.
How do you solve fractions step by step?
Add: 2 + 4 = 6
Multiple: 3 * the result of step No. 1 = 3 * 6 = 18
Subtract: 28 - the result of step No. 2 = 28 - 18 = 10
Add: the result of step No. 3 + 9 = 10 + 9 = 19
HOPE THIS HELPS!
the mayor of a city randomly chooses 4 of the 20 wards in her city and surveys all the voters in those selected wards.
Answers
The sample space is 4845.Therefore, option C is correct.
Given: The mayor of a city randomly chooses 4 of the 20 wards in her city and surveys all the voters in those selected wards.
Solution: We have a sample size of 4 wards out of a total of 20 wards. The mayor has chosen all the voters in those 4 wards. So, the sampling method used here is stratified sampling as the mayor chose the voters of specific 4 wards.
Sample space = Total number of ways to select 4 wards out of 20 wards= C(20,4) = 20!/(4!(20 - 4)!) = 4845
Hence, the sample space is 4845.Therefore, option C is correct.
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How to solve for j -10=║j-7║
Answers
Answer:
No solutions.
Step-by-step explanation:
-10 = 7
No solutions
BRAINLIEST
please help me ty
You can just do one question but I would appreciate both
Answers
Answer:
3. m1=82, m4=59. m5=98, m6=51, m7=79, m9=51, m10=129
4. m1=36, m2=90, m4=36, m5=90, m6=54, m7=126, m8=54, m9=126, m10=54, m11=36, m12=144, m13=36, m14=144
Step-by-step explanation:
3. m2=98, m3=23, m8=70
by supplementary angles, m1 + m2 = 180, since we know m2, we have m1 = 180 - 98 = 82. by corresponding angles we have m1 = m3 + m4, since we know m1 and m3, we have m4 = 82 - 23 = 59. again by supplementary angles, m3 + m4 + m5 = 180. we know m3 and m4 so m5 = 180 - 23 - 59 = 98. by alternate interior angles m4 = m7, so m7 = 59. by supplementary angles m6 + m7 + m8 = 180 so m6 = 180 - 70 - 59 = 51. by corresponding angles m6 = m9 so m9 = 51. by supplementary angles m9 + m10 = 180 so m10 = 129
4. m3=54
m2 is a right angle so m2=90. by complementary angles m3 + m4 = 90 so m4 = 36. by vertical angles m1=m4 so m1=36. by vertical angles m2=m5 and m3=m6 so m5=90 and m6=54. by corresponding angles m7 = m2+m1 so m7=126. by supplementary angles m7+m8 = m7+m10 = 180 so m8=10=54. by vertical angles m7=m9 so m9=126. by corresponding angles m1=m11 and m3=13 so m11 and m13 = 54. by supplementary angles m12+m13= m11+m14 = 180, so m12 = m14 = 144
Answer:
Step-by-step explanation:
Find the component form of the vector that translates P(-3,6) to P'(-4,8) .
Answers
Vector is the quantity that has magnitude as well as direction. The component form of the vector that translates P(-3,6) to P'(-4,8) is -i + 2j.
What is vector?A vector quantity has both magnitude and direction. It is added using parallelogram law of addition.
The sum of two vector quantities is always a vector quantity.
The sum of two vector quantities can vary from their scalar difference to their scalar addition.
The given points are P(-3,6) to P'(-4,8) .
The vector component form of two points (x₁,y₁) and (x₂,y₂) is given by ,
(x₂-x₁)i + (y₂-y₁)j
Therefore, The vector component form of given points are,
(-4-(-3))i + (8-6)j
-i + 2j.
Hence "The component form of the vector that translates P(-3,6) to P'(-4,8) -i + 2j.
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pls help me i kinda for got how to do this
Answers
Answer:
24 is x=2
Step-by-step explanation:
0.00000012+8*10^-7 scientific notation
Answers
Answer:
9.2*10^-7
Step-by-step explanation:
Move decimal until there is one number to the left. The number of times moved will be the exponent. If the decimal moves left the exponent is positive, if it moves left it is negative.
So 0.00000012=1.2*10^-7
1.2*10^-7 + 8*10^-7
9.2*10^-7
Two irrational numbers that equal to 2. What are they?
Answers
The two irrational numbers that sum up to 2 are (√5 - 2) and (4 - √5)
What are irrational numbers?Irrational numbers are numbers that cannot be expressed or written as a quotient of integers.
Example of irrational numbers are; √5 , √7, √3, etc
Irrational numbers that can add up to 2
(√5 - 2) + (4 - √5)
expand the bracket
√5-2 + 4-√5
collect like terms
√5 - √5 -2 + 4
-2 + 4
2
Hence, the two irrational numbers that sum up to 2 are (√5 - 2) and (4 - √5)
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What is the slope of the line that passes through the points shown in the table?
Answers
Hope this helped :)
Jason and Pam recently bought a house. They financed the house with a $250,000, 30-year mortgage with a nominal interest rate of 77% compounded monthly. Mortgage payments are made at the end of each month. What would be their house payment?
Answers
Answer:
Annual payument (PMT)= $1,663.19
Step-by-step explanation:
Giving the following information:
Loan (PV)= $250,000
Monthly interest rate (i)= 0.07/12= 0.005833
Number of periods (n)= 12*30= 360 months
To calculate the monthly payment, we need to use the following formula:
Annual payument (PMT)= (PV*i) / [1 - (1+i)^(-n)]
Annual payument (PMT)= (250,000*0.005833) / [1 - (1.005833^-360)]
Annual payument (PMT)= $1,663.19
someone please help- just look at the picture
Answers
Answer:
30
Step-by-step explanation:
18/60 =.3 x 100 = 30