Answer:
Step-by-step explanation:
Given that:
The equation of the damped vibrating spring is y" + by' +2y = 0
(a) To convert this 2nd order equation to a system of two first-order equations;
let y₁ = y
y'₁ = y' = y₂
So;
y'₂ = y"₁ = -2y₁ -by₂
Thus; the system of the two first-order equation is:
y₁' = y₂
y₂' = -2y₁ - by₂
(b)
The eigenvalue of the system in terms of b is:
[tex]\left|\begin{array}{cc}- \lambda &1&-2\ & -b- \lambda \end{array}\right|=0[/tex]
[tex]-\lambda(-b - \lambda) + 2 = 0 \ \\ \\\lambda^2 +\lambda b + 2 = 0[/tex]
[tex]\lambda = \dfrac{-b \pm \sqrt{b^2 - 8}}{2}[/tex]
[tex]\lambda_1 = \dfrac{-b + \sqrt{b^2 -8}}{2} ; \ \lambda _2 = \dfrac{-b - \sqrt{b^2 -8}}{2}[/tex]
(c)
Suppose [tex]b > 2\sqrt{2}[/tex], then λ₂ < 0 and λ₁ < 0. Thus, the node is stable at equilibrium.
(d)
From λ² + λb + 2 = 0
If b = 3; we get
[tex]\lambda^2 + 3\lambda + 2 = 0 \\ \\ (\lambda + 1) ( \lambda + 2 ) = 0\\ \\ \lambda = -1 \ or \ \lambda = -2 \\ \\[/tex]
Now, the eigenvector relating to λ = -1 be:
[tex]v = \left[\begin{array}{ccc}+1&1\\-2&-2\\\end{array}\right] \left[\begin{array}{c}v_1\\v_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]
[tex]\sim v = \left[\begin{array}{ccc}1&1\\0&0\\\end{array}\right] \left[\begin{array}{c}v_1\\v_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]
Let v₂ = 1, v₁ = -1
[tex]v = \left[\begin{array}{c}-1\\1\\\end{array}\right][/tex]
Let Eigenvector relating to λ = -2 be:
[tex]m = \left[\begin{array}{ccc}2&1\\-2&-1\\\end{array}\right] \left[\begin{array}{c}m_1\\m_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]
[tex]\sim v = \left[\begin{array}{ccc}2&1\\0&0\\\end{array}\right] \left[\begin{array}{c}m_1\\m_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]
Let m₂ = 1, m₁ = -1/2
[tex]m = \left[\begin{array}{c}-1/2 \\1\\\end{array}\right][/tex]
∴
[tex]\left[\begin{array}{c}y_1\\y_2\\\end{array}\right]= C_1 e^{-t} \left[\begin{array}{c}-1\\1\\\end{array}\right] + C_2e^{-2t} \left[\begin{array}{c}-1/2\\1\\\end{array}\right][/tex]
So as t → ∞
[tex]\mathbf{ \left[\begin{array}{c}y_1\\y_2\\\end{array}\right]= \left[\begin{array}{c}0\\0\\\end{array}\right] \ \ so \ stable \ at \ node \ \infty }[/tex]
Could someone please answer this I will give a brainless if answer correctly and explained
Answer:
SSS
Step-by-step explanation:
The SSS theorem is the side-side-side theorem. Basically, since all three sides from one triangle is congruent to the corresponding sides of the other triangle, then the triangles have to be congruent. There are other theorems such as the SAS theorem (side-angle-side), but since the question only give us the fact that the sides are congruent, its SSS. Sorry, I hope this is good enough. XD
Which equation represents the line whose slope is 2 and y-intercept is 6? a. 2y + 6x = 0 b. y = 6x + 2 c. y = 2x + 6 d. y + 2x = 6
Answer:
c. y=2x+6
Step-by-step explanation:
In slope intercept for (y=mx+b) m is the slope and b is the y-intercept.
How do you find the mean and standard deviation?
X + 7/8 = 23/8
Please help
Answer:
x=2
Hope I helped
Answer:
X=2
Step-by-step explanation:
Please mark brainliest.
Evan is making a table that will be created of the figure below. The table top is a triangle attached to a rectangle. To purchase the right amount of paint he needs to know the area of the table top He can only spend 10 on paint which is enough to cover 150ft of surfaces area. What is the max length of the base of the the rectangle he can build
Answer:
23 ft
Step-by-step explanation:
The computation of the maximum length of the base of the triangle is as follows:
Area of triangle is = 4 × 6 ÷ 2
= 12 ft²
Now the maximum area of the rectangle is
= 150 - 12
= 138 ft²
L = ?
W = 6 ft
Area of rectangle is = L × W
L × 6 = 138
L = 138 ÷ 6
= 23 ft
Help need it asap
8th grade math
Answer:
x = 7
Step-by-step explanation:
Given
2(2x + 1) = 30
4x + 2 = 30
4x = 30 - 2
4x = 28
x = 28/ 4
x = 7
Hope it helps :)❤
Answer:
x= 7
Step-by-step explanation:
2(2x+1) =30
2×2x+2×1=30
4x+2=30
4x=30-2
4x=28
x=28÷4
x=7
3 1/2 miles in 1/2 hour
Answer:
? 7 miles an hour i guess
Select the correct answer.
What is the slope of the linear relationship shown in this table of values?
a.-3
b.-2
c.2
d.3
Answer:
c
Step-by-step explanation:
0.4,0.7,1.0,1.3 what is the sequence rule
Answer:
3 plus 3, that is the sequence
In parallelogram MNPT, MZM - (6x + 10% and m2N - (5x + 10.5. How many
degrees is ZT?
brainlist asap, please help i need to boost my grade in math ♡´･ᴗ･`♡
♡︎♡︎♡︎♡︎♡︎♡︎♡︎♡︎♡︎♡︎♡︎♡︎
question: mona borrowed $6000 for college expenses. After 2 1/2 years she will have paid $750 in interest. What is the interest rate? ______
Answer:
5250
i hope this is right.
Answer:
5250
Step-by-step explanation:
person on my top got it ^
Math Problem, I will mark brainliet thank youu
A) 0.06 ✔✔.............
Answer:
A. [tex] 0.0 \bar6[/tex]
Step-by-step explanation:
[tex] \pink{ \huge \frac{1}{15}} = \orange{ 0.06666666....} \\ \\ \huge \purple{\frac{1}{15}} = \red{0.0 \bar6}[/tex]
Helpppp meeee fassttt pleaseeee
Answer:
y = -1/3x + 3 2/3
Step-by-step explanation:
step 1. find the slope of the given line
y2-y1/x2-x1 2 - 4/ 3 + 3 = - 1 / 3
take the opposite of that and there is your slope
step 2. plug in the values to the following equation using the given point:
y - y1 = m ( x - x1 ) y - 4 = - 1/3 ( x + 1 )
step 3. solve for y
y-4 = -1/3(x+1)
-1/3(x) and -1/3(1) ----> y-4 = -1/3x-1/3
y-4+4 = -1/3x-1/3+4 ----> y = -1/3x + 3 2/3
hope this helps. if you need more help lmk in the comments. i am in algebra two so you can trust me. Happy holidays
----------- are used to represent an unknown quantity in a mathematical expression.
HELP I WILL GIVE BRAINLEIST
Answer:
variables
Step-by-step explanation:
a
29
Solve each percent equation. Remember "of" means multiply and "is" means equal.
At your school, 74% of students bought their lunch at the cafeteria. There are
1250 students in the school. How many students bought their lunch?
925
students who brought their lunch. YUM!!
Answer
925 students brought their lunch.
Explanation
First let's rewrite the equation.
You know that "of" means multiply and "is" means equal.
Okay, so let's break it down.
The question says, "At your school, 74% of students brought their lunch at the cafeteria. There are 1250 students in the school."
This means that 74% of 1250 students brought their lunch at the cafeteria.
You are trying to find how many students brought their lunch. This will be x.
So the equation would be this.
74% × 1250 = x
74% is 0.74.
0.74 × 1250 = x
0.74 × 1250 = 925
So 925 students brought their lunch.
Marshall has $0.34 in dimes and pennies. He has 7 coins altogether. How many coins of each kind does he have?
Dimes are 0.10 each
Pennies are 0.01 each
You need 4 pennies to get the 0.04
Then you would have 3 dimes .
3 x 0.10 = 0.30
4 x 0.01 = 0.04
0.30 + 0.04 = 0.34
please i really really need help BRAIBLIST ASAP
5. A violinist practices one hour a day from Monday through Friday. How many hours must she practice on Saturday
order to have an average (antharctic meani ot two hours a day for the six day period?
K
Answer:
7 hours
Step-by-step explanation:
First find the total number of hours she practices based on the average of 2 hours a day for six days.
This means 6*2 = 12 , she practices a total of 12 hours.
Next find the total hours she already practiced. She practices 1 hour a day from Mon to Fri, which is 5 days, so 1*5 = 5 hours. This is the hours she practiced already.
To meet the mean, she needs to practice 12-5 = 7 hours more on Saturday.
please give thanks by clicking the red button :)
Answer:
must practice 7 hours on Saturday
Step-by-step explanation:
Let 'x' = number of hours on Saturday
(5 (hours) + x) / 6 (days) = 2
[tex]\frac{5+x}{6}[/tex] = [tex]\frac{2}{1}[/tex]
Cross-multiply to get 5 + x = 12
Can Someone help me!
Answer:< IJL= 63°
Step-by-step explanation:
as AB||CD, EF||GH, KLJI is a paralellogram, & the opposite angles of paralellogram are equal. So,< IJL= <IKL=63°
The tables shows the amount of sugar and the amount of water used in the fall. How many cups of sugar would be needed for 15 cups of water in the Fall Mixture?
Answer:
30
Step-by-step explanation:
it's x2 for the rest and 15x2 =30
I need help need it done in 20 minutes will give 20 points each
Answer:
Lets start by isolating for one variable In this case I chose y -7x-2y=14 -2y=14+7x y=-7 -3.5x (1)Next plug the y into the second equation to solve for x
6x+6(-7 -3.5x) =186x+(-42)+ (-21x) =18 6x-21x= 60 -15x=60x=-4Now we know x and we can plug into the original equation (1)
x= -4 y= -7 - 3.5× -4 y= 7Rate brainlist and drop a follow for future help
Answer:
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
6*x-6*y-(18)=0
STEP
1
:
Pulling out like terms
1.1 Pull out like factors :
6x - 6y - 18 = 6 • (x - y - 3)
Equation at the end of step
1
:
STEP
2
:
Equations which are never true:
2.1 Solve : 6 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Equation of a Straight Line
2.2 Solve x-y-3 = 0
Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).
Step-by-step explanation:
x= 1 y= 1
hope am right if i am, pls brainliest
Which is the completely factored form of 4x + 1Ox-6x?
2(x+3)(2x-1)
2x(x + 3)(2x- 1)
2x(x -3)(2x- 1)
2(x-3)(2x + 1)
Answer:
2(x+3)(2x-1)
Step-by-step explanation:
100 pts..[tex]ANSWER this question for your God's sake PLZ..[/tex]
Answer:
Step-by-step explanation:
a) [tex]5y+ab[/tex]
b) [tex]f^{3} -3\\[/tex]
c) [tex]6kq[/tex]
d) [tex]\frac{3xy}{2w} \\\\[/tex]
e) [tex]3x-4\sqrt{z} \\[/tex]
f) [tex]\frac{2p}{5q} \\\\[/tex]
[tex]a) 5y+ab5y+ab
b) \begin{gathered}f^{3} -3\\\end{gathered}
f
3
−3
c) 6kq6kq
d) \begin{gathered}\frac{3xy}{2w} \\\\\end{gathered}
2w
3xy
e) \begin{gathered}3x-4\sqrt{z} \\\end{gathered}
3x−4
z
f) \begin{gathered}\frac{2p}{5q} \\\\\end{gathered}
5q
2p
[/tex]
will give brainliest !
The state fair is a popular field trip destination. This year the senior class at High School A and
the senior class at High School B both planned trips there. The senior class at High School A
rented and filled 5 vans and 9 buses with 309 students. High School B rented and filled 12 vans
and 9 buses with 414 students. Every van had the same number of students in it as did the buses.
How many students can a van carry? How many students can a bus carry?
Answer:
15 students/van
26 students/bus
Painter has 7/8 liters of paint remaining each statue requires 1/20 how many statues can she paint
Answer:
1/20 liters per statue
so you divide 7/8 by 1/20
7/8*20/1
17.5 statues
but there cant be decimal statues,
so 17 statues is the answer
Answer:
17 statues.
Step-by-step explanation:
7/8 / 1/20
= 7/8 * 20
= 140/8
= 17.5.
please help me graph it!
Find the perimeter of the following shapes to 1 d.p take r =3.14
Answer:
The perimeter is 19.7
Step-by-step explanation:
The question is not complete and there is no attachment. Nevertheless, we can make do with the given information.
we can picture that the shape is a circle and we are required to find the perimeter/circumference of the circular shape
Given
The radius r= 3.142
Step two:
the perimeter/circumference is expressed as
C=2πr
C= 2*3.142*3.142
C= 19.7
Any help please I appreciate it thank you
Answer:
20 feet
Step-by-step explanation:
Parimeter is how far you would walk if you made a lap around the shape, walking on the edge lines like a balance beam.
To find perimeter, we need to add up all the sides.
Because it's a square each side has the same length.
There are 4 sides.
5 x 4 = 20
or 5 + 5 + 5 + 5 = 20
The perimeter is 20 feet
A runner burns 100 Calories for every 15 minutes spent running. How many hours of running would it take for a runner to burn 600 Calories?
GIVIVNG AWAY ALL MY POINTS FOR THIS QUESTION PLS ANSWER
Answer:
50 miles
Step-by-step explanation:
50 because 1 inch is 20 miles. There is two and a half inches.
