The given differential equation has characteristic equation
[tex]r^5 - 4r^4 + 4r^3 - r^2 + 4r - 4 = 0[/tex]
Solve for the roots [tex]r[/tex].
[tex]r^3 (r^2 - 4r + 4) - (r^2 - 4r + 4) = 0[/tex]
[tex](r^3 - 1) (r^2 - 4r + 4) = 0[/tex]
[tex](r^3 - 1) (r - 2)^2 = 0[/tex]
[tex]r^3 - 1 = 0 \text{ or } (r-2)^2=0[/tex]
The first case has the three cubic roots of 1 as its roots,
[tex]r^3 = 1 = 1e^{i0} \implies r = 1^{1/3} e^{i(0+2\pi k)/3} \text{ for } k\in\{0,1,2\} \\\\ \implies r = 1e^{i0} = 1 \text{ or } r = 1e^{i2\pi/3} = -\dfrac{1+i\sqrt3}2 \text{ or } r = 1e^{i4\pi/3} = -\dfrac{1-i\sqrt3}2[/tex]
while the other case has a repeated root of
[tex](r-2)^2 = 0 \implies r = 2[/tex]
Hence the characteristic solution to the ODE is
[tex]y_c = C_1 e^x + C_2 e^{-(1+i\sqrt3)/2\,x} + C_3 e^{-(1-i\sqrt3)/2\,x} + C_4e^{2x} + C_5xe^{2x}[/tex]
Using Euler's identity
[tex]e^{ix} = \cos(x) + i \sin(x)[/tex]
we can reduce the complex exponential terms to
[tex]e^{-(1\pm i\sqrt3)/2\,x} = e^{-x/2} \left(\cos\left(\dfrac{\sqrt3}2x\right) \pm i \sin\left(\dfrac{\sqrt3}2x\right)\right)[/tex]
and thus simplify [tex]y_c[/tex] to
[tex]y_c = C_1 e^x + C_2 e^{-x/2} \cos\left(\dfrac{\sqrt3}2x\right) + C_3 e^{-x/2} \sin\left(\dfrac{\sqrt3}2x\right) \\ ~~~~~~~~ + C_3 e^{-(1-i\sqrt3)/2\,x} + C_4e^{2x} + C_5xe^{2x}[/tex]
For the non-homogeneous ODE, consider the constant particular solution
[tex]y_p = A[/tex]
whose derivatives all vanish. Substituting this into the ODE gives
[tex]-4A = 69 \implies A = -\dfrac{69}4[/tex]
and so the general solution to the ODE is
[tex]y = -\dfrac{69}4 + C_1 e^x + C_2 e^{-x/2} \cos\left(\dfrac{\sqrt3}2x\right) + C_3 e^{-x/2} \sin\left(\dfrac{\sqrt3}2x\right) \\ ~~~~~~~~ + C_3 e^{-(1-i\sqrt3)/2\,x} + C_4e^{2x} + C_5xe^{2x}[/tex]
Jackson invested $4,200 in an account paying an interest rate of 9 1/2 compounded continuously. Julia invested $4,200 in an account paying an interest rate of 8 7/8 compounded quarterly. To the nearest hundredth of a year, how much longer would it take for Julia's money to double than for Jackson's money to double?
It would Julia 0.60 years more to double the initial investment.
What is future value?
Future value means the initial investment multiplied by 2 since the future value is meant to double.
The formula for future value of a continuously interest rate is provided below:
FV=PV*e^(rt)
FV=future value=$4,200*2=$8,400
PV=initial investment=$4,200
e=exponential constant=2.7182818
r=interest rate=9.5%
t=number of years it takes for the investment to double=unknown
$8,400=$4,200*2.7182818^(9.5%*t)
$8,400/$4,200=2.7182818^(0.095t)
2=2.7182818^0.095t
take log of both sides
ln(2)=0.095t* ln(2.7182818)
0.095t=ln(2)/ln(2.7182818)
0.095t=0.69314718781684800
t=0.69314718781684800/0.095
t=7.30 years
The future value when interest is compounded quarterly is shown thus:
FV=PV*(1+r/4)^(N*4)
FV=$8,400
PV=$4,200
r=8 7/8%
r=8.875%
N=the number of years it would take for the initial investment to double=unknown
$8,400=$4,200*(1+8.875%/4)^(N4)
$8,400=$4,200*(1+0.0221875)^(N4)
$8,400/$4,200=(1+0.0221875)^(N4)
2=(1+0.0221875)^(N4)
2=(1.0221875)^(N4)
take log of both sides
ln(2)=N4*ln(1.0221875)
N4=ln(2)/ln(1.0221875)
N4=31.5857423180125
N=31.5857423180125/4
N=7.90
Difference in years=7.90-7.30
difference in years=0.60 years
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Melissa is driving at a constant speed. She catches up to a minivan half a mile ahead of her that is
moving at 50 miles per hour in 50 seconds. How many miles per hour is the speed of the sports
car?
The speed of the sports car is 53.6 miles per hour
The key concept for solving such questions is the relationship between speed, distance and time.
The relationship between them is Distance = Speed x Time
Distance covered by Melissa = 0.5 miles + Distance covered by minivan
Let speed of Melissa be x miles per second
x. 50 = 0.5 + 50x50/3600
x .50 = 0.5 + 25/36
x = 1/100 + 1/72
x = 0.001 + 0.013
x = 0.014 miles per second
x = 3600 x 0.014 = 53.6 miles per hour
Thus the speed of the sports car is 53.6 miles per hour
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Convert the given rectangular
coordinates into polar form.
(−1,−2) → ([?], -[])
Round both to the nearest tenth and use radians.
The given coordinates to polar forms are:
[tex]\sqrt{5}[/tex]1.107How to find the polar formIn order to calculate it to polar forms we would have
We have these equations
r² = x² + y²
tanθ = y/x
From here we would have
r² = (-1)² + (-2)
r² = 1 + 4
r = √5
We have tanθ = -2 / -1
= we have θ = tan⁻¹(2)
= 1.107
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Two trees are 120 m apart. From the point halfway between them, the angle of elevation to the top of the trees is 36 and 52. How much taller is one tree than the other.
One tree is 31.044 m taller than the other one in height.
Given Information and Formula Used:
The distance between the trees, BC (from the figure) = 120 m
Elevation of angles to the top of the trees,
∠AMB = 52°
∠DMC = 36°
In a right angled triangle,
tan x = Perpendicular/Height
Here, x is the angle opposite to the Perpendicular.
Calculating the Height Difference:
Let's compute the height of the taller tree first.
In ΔAMB,
tan 52° = AB / BM
Now, since M is the point halfway between the trees,
BM = CM = BC/2
BM = CM = 60 m
⇒ 1.2799 = AB / 60
AB = 1.2799 × 60
Thus, the height of the taller tree, AB = 76.794 m
Now, we will compute the height of the smaller tree.
In ΔDCM,
tan 36° = DC / CM
⇒ 0.7625 = DC / 60
DC = 0.7625 × 60
Thus, the height of the smaller tree, DC = 45.75 m
The difference in the heights of the trees, AP = AB - DC
AP = (76.794 - 45.75)m
AP = 31.044m
Hence one tree is 31.044m taller in height than the other.
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What are the solutions to 4x^3 + 16x^2 + 28x = 0
Step-by-step explanation:
the third option is the answer.
Help me please I’m not the smartest
Need help with this one
Answer:
The answer is a.2x/9Step-by-step explanation:
[tex] \frac{x {}^{2} }{3y} \div \frac{3x}{2y} \\ \frac{x {}^{2} }{3y} \times \frac{2y}{3x} \\ \frac{x {}^{2} }{3} \times \frac{2}{3x} \\ \frac{x}{3} \times \frac{2}{3} = \frac{2x}{9} [/tex]
what is the inverse of the following function? 50 points
The inverse of the given function [tex]f(x)=\sqrt[5]{x^3}[/tex] is [tex]f^{-1}(x)=x^{\frac{5}{3}}[/tex]
Inverse of a functionThe given function is:
[tex]f(x)=\sqrt[5]{x^3}[/tex]
This function can be written in exponent form as:
[tex]f(x)=x^\frac{3}{5}[/tex]
Make x the subject of the formula:
[tex]x=[f(x)]^\frac{5}{3}[/tex]
Let x be replaced by [tex]f^{-1}(x)[/tex] and f(x) be replaced by x
The inverse function therefore becomes:
[tex]f^{-1}(x)=x^{\frac{5}{3}}\\\\ f^{-1}(x)=\sqrt[3]{5}[/tex]
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help me please
What is the midpoint of the horizontal line segment graphed below?
A. (4,3)
B. (8,6)
C. (8,3)
D. (4, 6)
-10
10
(-2,3)
$10
(10, 3)
10
Answer: A
Step-by-step explanation:
39 kg Rice of taken out of a bag of rice. If 8/22 of the rice is left in the beg then find the quantity of rice the bag contained in the beginning
Answer:
429/7 kg or 61 2/7 kg
Step-by-step explanation:
I'll answer according to how I understand this problem, but I'm not sure this is correct.
"39 kg of rice is taken out of bag of rice. If 8/22 of the rice is left, what was the original weight of the rice in the bag?"
8/22 is a strange number to work with because it could have been given as 4/11. This is another reason I'm not too sure about this problem.
8/22 = 4/11
If 4/11 is left, then since 11/11 - 4/11 = 7/11, 7/11 of the rice remains.
7/11 x = 39
x = 39 × 11/7
x = 429/7
x = 61 2/7
Graph the function f(x) = 2√x +3.
The graph is shown in the attached image.
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Proving a uadrilateral is a Parallelogram
Assignment Active
Determining Side Lengths in a Parallelogram
In quadrilateral ABCD, AD 1, BC.
3x+7
D
A
B
5x-9
What must the length of segment AD be for the
quadrilateral to be a parallelogram?
O 8 units
O 16 units
O 31 units
O 62 units
Answer:
31 units
Step-by-step explanation:
If a quadrilateral has a pair of sides that is both parallel and congruent, then the quadrilateral is a parallelogram.
Sides AD and BC are given as parallel.
If they are also congruent, then quadrilateral ABCD is a parallelogram.
3x + 7 = 5x - 9
-2x = -16
x = 8
AD = 3x + 7
AD = 3(8) + 7
AD = 31
Answer: 31 units
factor 18x^(2)+39x-15
Answer:
[tex]3(2x + 5)(3x - 1)[/tex]
Step-by-step explanation:
Hello!
We can first factor out the greatest common factor between the coefficients: 3.
[tex]f(x) = 18x^2 + 39x - 15[/tex][tex]f(x) = 3(6x^2 + 13x - 5)[/tex]Now, let's work with what we have inside the brackets.
Standard form of a quadratic: [tex]ax^2 + bx+ c = 0[/tex]
Given our equation: [tex]6x^2 + 13x - 5[/tex]
a = 6b = 13c = -5We need to find two numbers that add up to b (13) and multiply to ac (-30).
The two numbers are 15 and -2. Expand 13x to 15x and -2x and factor by grouping.
Factor by Grouping[tex]6x^2 + 13x - 5[/tex][tex]6x^2 -2x + 15x - 5[/tex][tex]2x(3x- 1) +5(3x - 1)[/tex][tex](2x +5)(3x - 1)[/tex]
Now, we simply add the other factor, 3, to the final factored form.
Factored Solution: [tex]3(2x + 5)(3x - 1)[/tex]
help guys please. I
Step-by-step explanation:
you need to use the ruler to measure the lengths of it, like place the ruler from A to B and look at the measurement of it
the set of all months that contain less than 31 years
The set of all months that contains less than 31 days is; {February, April, June, September, November}.
What is the set of all months that contain less than 31 days?It follows from the task content that the set whose elements are to be determined is characterized by containing less than 31 days.
On this note, it follows that the elements of the set are; February (28/29 days), April (30 days), September (30 days), June (30 days) and November (30 days).
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If a person rows to his favorite fishing spot 21 miles downstream in the same amount of time that he rows 7 miles
upstream and if the current is 7 mph, find how long it takes him to cover 28 miles.
28 miles upstream requires 28/7 = 4 hours.
28 miles downstream requires 28/21 = 4/3 hours, or 1 hour and 20 minutes.
How long does it take him to cover 28 miles?Apply the distance formula:
d = rt
where
d is distance
r is rate or speed
t is time
21 = t*(r+7) = rt+ 7t
7 = t *(r-7) = rt - 7t
28 = 2rt
14 = rt
21 = rt + 7t
21 = 14 + 7t
7 = 7t
t=1
21 = (r+7)*1 = r+7
r = 14
The speed of the boat is 14 mph. With the current (downstream), the speed is 14+7 = 21 mph, so 21 miles are traveled in 1 hour. Against the current (upstream), the speed is 14-7 = 7 mph, which means 7 miles are traveled in the same 1-hour time frame.
28 miles downstream requires 28/21 = 4/3 hours, or 1 hour and 20 minutes.
28 miles upstream requires 28/7 = 4 hours.
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Find the sum of the arithmetic series given a1=45, an=85, and n=. 5
The sum of the arithmetic series is 325
How to determine the sum?The given parameters are
a1 = 45
an = 85
n = 5
The formula for the sum of arithmetic series is:
[tex]S_n = \frac n2 * (a_1 + a_n)[/tex]
Substitute the values of a1, an and n in the above equation
[tex]S_5 = \frac 52 * (45 + 85)[/tex]
Add 45 and 85
[tex]S_5 = \frac 52 * 130[/tex]
Divide 130 by 2
[tex]S_5 = 5 * 65[/tex]
Multiply
[tex]S_5 = 325[/tex]
Hence, the sum of the arithmetic series is 325
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Give the name (monomial,
binomial, trinomial etc.) and
degree of the polynomial.
3x4
Name? [?]
Degree? [ ]
Answer:
monopoly, order 1
Step-by-step explanation:
because the x has just power of one that is why is monopoly and in order 1
Graph the image of the polygon after a reflection in the line y = t.
Answer:
see the attached
Step-by-step explanation:
Reflection over a line moves each point so that the segment between it and its image has the reflection line as its perpendicular bisector.
Reflection over y=xThe line of reflection y=x has the effect of swapping the x- and y-coordinates:
(x, y) ⇒ (y, x) . . . . . reflection in y=x
For example, point C(2, 3) moves to point C'(3, 2).
The attachment shows the reflected figure.
which of the following functions is graphed below?
The graphed piecewise function is the one in option B.
Which of the given functions is the graphed one?
On the graph, we can see that on the lowest part we have a closed dot at x = 1
And the above part has a open dot at x = 1.
Then the piecewise function is of the form:
y = f(x) for x ≤ 1y = g(x) for x > 1Such that the above part seems to be quadratic, and the part below seems to have a larger degree.
With that, we can conclude that the correct option is:
y = x^2 + 4, for x > 1y = x^3 - 2, for x ≤ 1Which is option b.
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Which person is vulnerable to identity theft
The person which is more vulnerable to identity theft include:
The elderlyCollege studentsChildrenWhat is Identity theft?This involves using someone's personal information to commit fraud and other forms of crime.
The set of people mentioned above are usually naive and innocent thereby making them more vulnerable to identity theft.
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Find the difference: -2-6
Identify a pair of factors of −35 that has a sum of −2.
Answer:
-7 and 5
Step-by-step explanation:
If you multiply -7 and 5, you get -35
Add them, and you get -2.
Jed wants to prove that his test scores are greatly improving. He makes the graph shown here.
Explain why someone may think this graph is misleading.
Someone may think this graph is misleading because it does not start from the origin
How to determine the reason?As a general rule, graphs are to begin from the origin
The origin of a graph is
(x, y) = (0, 0)
From the given graph, the origin is
(x, y) = (0, 60)
This means that someone may think this graph is misleading because it does not start from the origin
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Someone help quickly pls :(
Assignment
Practice finding solutions to systems of equations using
substitution.
The value of x in this system of equations is 1.
3x + y = 9
y=-4x+10
1. Substitute the value of y in the first equation:
2. Combine like terms:
3. Apply the subtraction property of equality:
4. Apply the division property of equality:
What is the value of y?
y=
3x + (-4x+10) = 9
-x+10=9
-x=-1
x=1
Answer:
y = 6
Step-by-step explanation:
Solving system of linear equation using substitution method:3x + y = 9 ---------------(I)
y = -4x + 10 -----------------(II)
Substitute y = -4x + 10 in equation (I)
3x - 4x + 10 = 9
Combine like terms,
-x + 10 = 9
Subtract 10 from both sides. (subtraction property of equality)
-x = 9 - 10
-x = -1
Divide both sides by (-1). {Division property of equality}
[tex]\sf \dfrac{-x}{-1}=\dfrac{-1}{-1}\\\\ \boxed{x = 1}[/tex]
Now, substitute x = 1 in equation (II) and find the value of y.
y = -4*1 + 10
= -4 + 10
[tex]\sf \boxed{\bf y = 6}[/tex]
Rafael earns $35 more per week than Andrew. If their weekly salaries total $715, what amount does each earn?
Answer:
Rafael earns $375
Andrew earns $340
Step-by-step explanation:
Here, we will set up a system of equations where r represents Rafael and a represents andrew.
r = a + 35
r + a = 715
Next, we substitute a + 35 for r in the second equation.
(a + 35) + a = 715 (combine like terms)
2a + 35 = 715 (subtract 35 from both sides)
2a = 680 (divide both sides by 2)
a = 340
To solve for r, we can plug 340 for a into either equation.
r = (340) + 35
r = 375
We can double check by plugging both values into the other equation.
375 + 340 = 715 (yes, it works!)
Brainliest, please :)
Find the limit
lim x->0 (4x/tan2x)
Answer:
Step-by-step explanation:
hello : here is a solution
PLEASE FAST!!! Explain why, given a large unit A and a small unit B, it is wise to leave the conversion factor in fractional form if the decimal form goes on forever.
Numbers whose decimals form goes on forever are best represented as fractions because they are non-terminating decimals
How to determine the reasons?The decimal numbers are represented as:
A and B
Where
A > B
Numbers whose decimals form goes on forever are non-terminating numbers, and the complete form cannot be represented using decimals except with the sign .......
Because non-terminating decimals have no end, it is best to represent them as fractions
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Pls help me i'm stuck
Answer: [tex]m\angle BAM=30^{\circ}[/tex]
Step-by-step explanation:
[tex]AD=x\sqrt{3}[/tex] (Pythagorean theorem)
[tex]m\angle ACD=60^{\circ}[/tex] (30-60-90 triangle)
[tex]m\angle MCA=30^{\circ}[/tex] (angle subtraction)
[tex]m\angle MAC=30^{\circ}[/tex] (base angles theorem)
[tex]m\angle CAD=30^{\circ}[/tex] (30-60-90 triangle)
[tex]m\angle BAM=30^{\circ}[/tex] (angle subtraction)
A total of $5000 is invested: part at 7% and the remainder at 12%. How much is invested at each rate if the annual interest is $400?
The amount invested in the account that yields 7% interest is $4000.
The amount invested in the account that yields 12% interest is $1000.
What are the linear equations that represent the question?a + b = 5000 equation 1
0.07a + 0.12b = 400 equation 2
Where:
a = amount invested in the account that yields 7% interest.
b = amount invested in the account that yields 12% interest.
How much is invested at each rate?
Multiply equation 1 by 0.07
0.07a + 0.07b = 350 equation 3
Subtract equation 3 from equation 2
0.05b = 50
b = 50 / 0.05
b = 1000
Subtract 1000 from 5000: 5000 - 1000 = 4000
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