Consider the following initial-value problem. dy at² dy 4 dt - 5y = 0, y(1) = 0, y'(1) = 7 - Find all the roots of the auxiliary equation. (Enter your answer as a comma-separated list.) Solve the giv
Answers
The solution to the given initial-value problem is: y(t) = (-1/6)e^(5t - 6) + (1/6)e^(4 - t) and the roots of the auxiliary equation are m1 = 5 and m2 = -1. This can be answered by the concept of differential equation.
Consider the given differential equation:
(dy/dt) - 4(dy/dt²) - 5y = 0, y(1) = 0, y'(1) = 7.
First, let's find the roots of the auxiliary equation. The auxiliary equation for this problem is:
m² - 4m - 5 = 0.
Factoring this equation, we get:
(m - 5)(m + 1) = 0.
So the roots of the auxiliary equation are m1 = 5 and m2 = -1.
Now that we have the roots, we can write the general solution for the given differential equation as:
y(t) = c1 × e^(5t) + c2 × e^(-t).
To solve for the constants c1 and c2, we can use the initial conditions provided:
1. y(1) = 0:
0 = c1 × e⁵ + c2 × e^(-1).
2. y'(1) = 7:
7 = 5c1 × e⁵ - c2 × e^(-1).
Solving this system of equations, we obtain c1 = -1/6 × e^(-6) and c2 = 1/6 * e⁴.
So, the solution to the given initial-value problem is:
y(t) = (-1/6)e^(5t - 6) + (1/6)e^(4 - t).
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Related Questions
pleasee help me
2t+ 1 = 3t-1 + 3
3 4
Answers
Answer:
t = -1
Step-by-step explanation:
\(2t + 1 = 3t - 1 + 3\)
Collect like terms and simplify
\(2t - 3t = - 1 + 3 - 1 \\ - t = 1\)
Divide through by -1
\( \frac{ - t}{ - 1} = \frac{ 1}{ - 1} \\ t = - 1\)
Answer:
t = - 2
Step-by-step explanation:
\(\frac{2t + 1}{3} = \frac{(3t-1) + 3}{4}\)
First we need to cross multiply to get rid of the division to make it easier to solve.
Multiply 2t+1 by 4 and (3t-1) + 3 by 3
4 ( 2t + 1 ) = 3 ( ( 3t - 1 ) + 3 ) On the right we need to combine terms.
4 ( 2t + 1 ) = 3 ( 3t - 1 + 3 )
4 ( 2t + 1 ) = 3 ( 3t + 2 ) Multiply
4 * 2t + 4 * 1 = 3 * 3t + 3 * 2 Remember order of operations. Multiplication comes before addition and subtraction.
4 * 2t + 4 * 1 = 3 * 3t + 3 * 2
8t + 4 = 9t + 6 Subtract 8t from each side
8t - 8t + 4 = 9t - 8t + 6
4 = t + 6 Subtract 6 from each side
4 - 6 = t + 6 - 6
4 - 6 = t
- 2 = t
Jason has three more than four times the amount of money Jeff has together they have $72 how much money does each person have
Answers
hello
determine the amount each of them have, let's write an equation to represent thier total amount
let x represent the amount Jason have
let y represent the amount Jeff has
now, we know that Jason has 3 more than 4 times the amount Jeff has
x = 4y + 3 .... equation 1
x + y = 72 .... equation 2
from equation 2,
make x the subject of formula
\(\begin{gathered} x+y=72 \\ x=72-y \end{gathered}\)x = 72 - y .....equation 3
put equation 3 into equation 1
x = 4y + 3
72 - y = 4y + 3
solve for y
\(\begin{gathered} 72-y=4y+3 \\ \text{collect like terms} \\ 4y+y=72-3 \\ 5y=69 \\ y=\frac{69}{5} \\ y=13.8 \end{gathered}\)we know y = 13.8
we can simply substitute the value into equation 2 and solve for x
\(\begin{gathered} x+y=72 \\ y=13.8 \\ x+13.8=72 \\ x=72-13.8 \\ x=58.2 \end{gathered}\)therefore, Jason has $58.2 and Jeff has $13.8
If α and β are the zeroes of the polynomial x2 – 7x + 10 , find the value of α3 + β3
Answers
If α and β are the zeroes of the polynomial x² - 7x + 10, then the value of α³ + β³ is = 133.
We know that "α" and "β" are the zeroes of the polynomial x² - 7x + 10,
we factor the polynomial by splitting the middle term ,
We get;
⇒ x² - 7x + 10 ⇒ x² - 2x - 5x + 10,
⇒ x(x - 2) -5(x - 2) ⇒ (x - 5)(x - 2).
So, we get that α = 5 and β = 2;
In order to find the value of the expression α³ + β³, we substitute, the values of α and β in the expression,
We get;
⇒ 5³ + 2³
⇒ 125 + 8
⇒ 133.
Therefore , the value of expression α³ + β³ is 133.
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The given question is incomplete, the complete question is
If α and β are the zeroes of the polynomial x² - 7x + 10 , find the value of α³+β³.
Find the measure of the indicated angle to the nearest degree.
Answers
Answer:
? ≈ 20°
Step-by-step explanation:
using the cosine ratio in the right triangle
cos ? = \(\frac{adjacent}{hypotenuse}\) = \(\frac{16}{17}\) , then
? = \(cos^{-1}\) ( \(\frac{16}{17}\) ) ≈ 20° ( to the nearest degree )
Guided Practice
Give a reason to justify the equation.
3y+ (-5y) = [3+ (-5)]y
A. Commutative Property of Addition
B. Associative Property of Addition
C. Distributive Property
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– 6 x + 4 < 29 answer?
Answers
Answer:
its 847>×+18 enjoy hope it work
what is the answer for 12*12 ?
Answers
Answer:
144
Step-by-step explanation:
Hope this helps! God bless you!
Solve for n: 2n = n + 3 O A. 1 O O c. 3 3 OD. 5 E 6
Answers
Step 1: Problem
2n = n + 3
Step 2: Concept
Collect like terms and find the value of x.
2n = n + 3
2n - n = 3
n = 3
Step 3: Final answer
n = 3
a class of 20 students is divided into two groups of 10 and then each group is lined up in a particular order. how many ways are there to divide the class and line up the groups?
Answers
There are ²⁰С₁₀ × 10! × 10! ways to divide the class and line up the groups.
Number of students in the class = 20
Selection of group one = ²⁰С₁₀ ways
Selection of group two = 1 way
Number of ways to arrange 10 students in a group = 10!
So, Total number of ways = ²⁰С₁₀ × 10! × 10! ways.
Hence there are ²⁰С₁₀ × 10! × 10! ways to divide the class and line up the groups.
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Aman borrowed r. 30,000 for 2 year at 10% P. A compounded annually. He paid only half of the principle at the end of 2 year. He paid the remaining principle and interet at the ame rate at the end of the next 2 year. How much amount did he pay. Atlat to clear the debt?
Answers
The Aman paid a total of Rs. 25,773, including Rs. 10,773 in compound interest, to settle the obligation.
What is meant by compound interest ?When you earn interest on your interest earnings as well as the money you have saved, this is known as compound interest. As an illustration, if you put $1,000 in an account that offers 1% yearly interest, you will receive $10 in interest after a year. Due to compound interest, Year Two would see interest payments of $10.10 on a total of $1,010 in principal and interest. The amount of additions increases the compound interest rate. Therefore, if you borrow Rs. 100. If you don't make any repayments, the amount will be Rs 110 after a year at 10% compound interest. Due to the addition of the interest-on-interest, the amount in the second year will be Rs 121.The amount of the loan = Rs.30,000
Loan period = 4 years
Annual percentage rate = 10%
Compounding period = Annual
Amount after two years = P( 1 + r/100)^n
= Rs. 30,000 x (1 + 0.1)^2
Rs. 36,300 (Rs. 30,000 x 1.21)
Payment after two years = Rs.15,000 (Rs. 30,000 x 50%)
Balance = Rs. 21,300 (Rs. 36,300 - Rs. 15,000)
Amount after another two years = Rs. 21,300 x (1 + 0.1)^2
= Rs. 25,773 ($21,300 x 1.21)
Total payments = Rs. 40,773 (Rs. 25,773 + Rs. 15,000)
Compounded interest = Rs. 10,773 (Rs. 40,773 - Rs. 30,000)
Thus, to settle the loan, lastly, Aman paid Rs. 21,300.
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If 9^x / 9^3 = 9^5 what is x
Answers
Answer:
since , 9^x / 9^3 = 9^5
and since there is a law for exponents that says if the bases of the exponent are equal and divided by each other we subtract the exponents
, so 9^x-3 = 9^5
then x-3 =5 , therefore the final answer will be X= 8 .
PLEASE HELP ASAP
What is the area of the real object that the scale drawing models?
A. 20 sq cm
B. 180 sq cm
C. 120 sq cm
D. 60 sq cm
Answers
Answer:
B. 180 sq cm
Step-by-step explanation:
if the area of the scale drawing is 20 cm then the width must be 5cm and the length is 4cm.
Since the scale factor is 1:3 you multiply the side lenghts by 3. So the width of the real object is 15 cm and the length is 12 cm.
Multiply L x W to get area, 15 x 12 = 180
A square has sides of length 6/12 inches. Area of length times width.
What is the area of the square in square inches?
Answers
The area of the square is 1/4inches² in square inches
What is area of squareThe area of a square is calculated by multiplying its two sides, that is area = s × s, where, 's' is one side of the square.
The square has side of length = 6/12
this can be simplified as 1/2
so
area of the square = (1/2 × 1/2) inches ²
area of the square = 1/4inches²
Thus, the area of the square is calculated using area = s × s, as 1/4inches²
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Help me please......
Answers
Answer:
A is the correct answer
Vail Ski Shop received a $1,205 invoice dated July 12 with 2/10, 1/15, n/60 terms. On July 26, Vail sent a $489 partial payment.
a. What credit should Vail receive?
b.What is Vail's outstanding balance
Answers
Outstanding balance = Invoice amount - CreditInvoice amount = $1,205Credit = $611.25Outstanding balance = $1,205 - $611.25Outstanding balance = $593.75Therefore, Vail's outstanding balance is $593.75.
a. To determine the credit that Vail should receive, we need to use the formula for partial payment: Payment / (1 - Discount%)The calculation for the credit that Vail should receive is:Discount% = 2/10 = 0.2Credit = 489 / (1 - 0.2)Credit = 489 / 0.8Credit = $611.25The credit that Vail should receive is $611.25.b.
To calculate Vail's outstanding balance, we need to subtract the credit from the invoice amount. The calculation is:Outstanding balance = Invoice amount - CreditInvoice amount = $1,205Credit = $611.25Outstanding balance = $1,205 - $611.25Outstanding balance = $593.75Therefore, Vail's outstanding balance is $593.75.
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Solve for u.
-2|3u| +5 > -5
Answers
The value of u is -5/3 < u < 5/3
We need to calculate the value of u
-2|3u| +5 > -5
-2(3u) + 5 > -5
-6u + 5 > -5
-6u > -10
6u < 10
3u < 5
u < 5/3
-2(-3u) + 5 > -5
6u + 5 > -5
6u > -10
3u > -5
u > -5/3
Therefore, the value of u is -5/3 < u < 5/3
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Ed's new car has a value of $23,500, but it is expected to depreciate at a rate of 7.5% per year. If you were to write an exponential decay function to represent this situation, what would the decay factor be?
Answers
Answer:
0.925 or 92.5%
Step-by-step explanation:
y=a(1-b)^x
decay factor is 1-b <- and the part we care about
b is 7.5%=0.075
1-0.075
=0.925
I'm a little rusty on exponential rates, but I hope this helps!
The decay factor for the car will be equal to 0.925 or 92.5%
What is an exponential expression?Powers can simply be expressed in concise form using exponential expressions. The exponent shows how many times the base has been multiplied. Since 2 is the "base" and 5 is the "exponent," it can be written as 2 x 2x 2 x 2=25 for the number 32. This phrase should be understood as "two to the fifth power."
Given that Ed's new car has a value of $23,500, it is expected to depreciate at a rate of 7.5% per year.
The decay factor will be calculated as,
\(y=a(1-b)^x\)
b is 7.5%=0.075
Decay factor = 1-0.075
Decay factor = 0.925
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Which expression is equivalent to (x-4y/x-9y5)-2
•y8/x10
•x5/y7
•x5/y4
•x/y7
Answers
Answer:
y8/x10
Step-by-step explanation:
Answer:
a
Step-by-step explanation:
cause math
The Root cause analysis uses one of the following techniques: o Rule of 72 o Marginal Analysis o Bayesian Thinking o Ishikawa diagram
Answers
The Root Cause Analysis technique used to identify the underlying causes of a problem is the Ishikawa diagram. It is a graphical tool also known as the Fishbone diagram or Cause and Effect diagram. The other techniques mentioned, such as the Rule of 72, Marginal Analysis, and Bayesian Thinking, are not specifically associated with Root Cause Analysis.
Root Cause Analysis is a systematic approach used to identify the fundamental reasons or factors that contribute to a problem or an undesirable outcome. It aims to go beyond addressing symptoms and focuses on understanding and resolving the root causes. The Ishikawa diagram is a commonly used technique in Root Cause Analysis. It visually displays the potential causes of a problem by organizing them into different categories, such as people, process, equipment, materials, and environment. This diagram helps to identify possible causes and facilitates the investigation of relationships between different factors. On the other hand, the Rule of 72 is a mathematical formula used to estimate the doubling time or the time it takes for an investment or value to double based on compound interest. Marginal Analysis is an economic concept that involves examining the additional costs and benefits associated with producing or consuming one more unit of a good or service. Bayesian Thinking is a statistical approach that combines prior knowledge or beliefs with observed data to update and refine probability estimates. In the context of Root Cause Analysis, the Ishikawa diagram is the technique commonly used to visually analyze and identify the root causes of a problem.
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Answer choices:-the function is never increasing. -the function is sometimes increasing.Increasing interval:__-the function is always increasing.Increasing interval:__
Answers
Given:
The objective is to write the interval notataion of increasing, decreasing and constant intervals.
1)
In the given graph, the function is increasing sometimes.
The increasing interval can be written as,
\(\text{Increasing interval =\lbrack{}1,}\infty)\)2)
In the given graph, the function is decreasing sometimes.
The decreasing interval can be written as,
\(\text{Increasing interval =\lbrack{}1,}\infty)\)activities g, p, and r are the immediate predecessors for activity w. if the earliest finish times for the three are 12, 15, and 10, then the earliest start time for w a. is 10. b. is 12. c. is 15. d. cannot be determined.
Answers
The earliest start time for activity w can be determined by taking the maximum of the earliest finish times of its immediate predecessors, which are activities g, p, and r. In this case, the maximum is 15. Therefore, the correct answer is c. is 15.
What is the equation of the line that passes through the points (-6, -10) and (8, 3)? Write your answer in slope-intercept form.
Answers
Answer:
y = 13/14 x - 31/7
Step-by-step explanation:
slope = (-10-3)/(-6-8)
slope = -13/-14
slope = 13/14
y-3 = 13/14 (x - 8)
y - 3 = 13/14 x - 52/7
y = 13/14 x - 52/7 + 3
y = 13/14 x + (-52 + 21)/7
y = 13/14 x - 31/7
I need to show my work please help and I’m in 7th grade i don’t know why it says highschool
Answers
Answer:
green sugar...................
because ....
1 1/2 =3/4....
factor (3c + 2)^2 - (3c - 2)^2
Answers
Hi there! Your answe is 24c
Please see an explanation below for a clear understanding to your problem.
Any questions about my explanation and answer can be asked through comments! :)
Step-by-step explanation:
\( \large \boxed{ {x}^{2} - {y}^{2} = (x + y)( x - y)}\)
The formula above is what we will be using. This is called "Difference of Two Squares".
Notice how the these expressions are similar to the formula's structure.
\( \large{ {(3c + 2)}^{2} - {(3c - 2)}^{2} }\)
Notice how there are minus sign, squared expression which we can factor by using the formula above. Therefore -
\( \large{ {x}^{2} = {(3c + 2)}^{2} } \\ \large{x = 3c + 2}\)
Therefore, our y would be 3c-2.
Then proceed with the formula.
\( \large{ {(3c + 2)}^{2} - {(3c - 2)}^{2} = (3c + 2 + 3c - 2)(3c + 2 - (3c - 2))}\)
Simplify.
\( \large{ (6c)(3c + 2 - 3c + 2)} \\ 6c(4) \\ 24c\)
Can you explain to me how to solve this?????
√19x^5
Answers
The final step when solving the given math problem is:
Take the fifth root of both sides: x = \(((y^2)/19)^(^1^/^5)\)
How to solveTo solve √19x^5 for x, follow these steps:
Isolate the square root term: \(\sqrt{19x^5}\) = y (Let y be the other side of the equation)
Square both sides: \((y^2) = 19x^5\)
Divide both sides by 19: \((y^2)/19 = x^5\)
Take the fifth root of both sides: x = \(((y^2)/19)^(^1^/^5^)\)
The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol √ and can be found using mathematical operations.
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Which one of these is x please help
Answers
Answer:
1/2
Step-by-step explanation:
The smallest subset of rational numbers is integers.
True
False
Answers
Answer:
False
Step-by-step explanation:
the question is in the photo
Answers
Answer:
d is vertical so its 70 b c and a combined is also vertical to e so we have 110
Step-by-step explanation:
Convert repeating decimal to a rational number (limit repeating decimals to
thousandths).
21.) 0.6=
22.) 1.25 =
23.) 0.45
Answers
21) You most likely mean the number 0.666…. If x = 0.666…, then
10x = 6.666…
10x - x = 6.666… - 0.666…
9x = 6
x = 6/2 = 2/3
22) You either mean 1.2555… (only 5 repeats) or 1.252525… (both 2 and 5 repeat).
If x = 1.2555…, then
10x = 12.555…
100x = 125.555…
100x - 10x = 125.555… - 12.555…
90x = 113
x = 113/90
Otherwise, if x = 1.252525…, then
100x = 125.252525…
100x - x = 125.252525… - 1.252525…
99x = 124
x = 124/99
23) You either mean 0.4555… or 0.454545….
If x = 0.4555…, then
10x = 4.555…
100x = 45.555…
100x - 10x = 45.555… - 4.555…
90x = 41
x = 41/90
Otherwise, if x = 0.454545…, then
100x = 45.454545…
100x - x = 45.454545… - 0.454545…
99x = 45
x = 45/99 = 5/11
What triangle is this ??? PLEASEE helpppp !!!!!!!! Will mark Brianliest !!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
isoscoles right
Step-by-step explanation: