Solve the differential equation by variation of parameters. (Use C1 and C2 as arbitrary constants.) 4y'' − 8y' + 8y = e^(x) sec x Solve step by step and please give correct answer!
Answers
The particular solution is y_p = (-1/36 e^(-2x) + Bx + C) e^x cos(x) +C.
How to solve differential equation?
To solve this equation, we find characteristic equation:
4r^2 - 8r + 8 = 0
Simplifying, we get:
2r^2 - 4r + 4 = 0
r^2 - 2r + 2 = 0
Using the quadratic formula, we get:
r = (2 ± sqrt(2)i)/2
The general solution of the homogeneous equation is:
y_h = e^x (C1 cos(sqrt(2)x/2) + C2 sin(sqrt(2)x/2))
To find the particular solution, we assume the following form for y_p:
y_p = u(x) e^x cos(x) + v(x) e^x sin(x)
We then calculate y_p', y_p'' and substitute these into the differential equation to obtain:
4(u'' + 2u' + v'' + 2v') e^x cos(x) + 4(u' - v) e^x sin(x) = e^x sec(x)
Equating the coefficients of cos(x) and sin(x) :
4u'' + 8u' + 4v'' + 8v' = 1 (equation 1)
4u' - 4v = 0 (equation 2)
Simplifying equation 2, we get:
u' = v
Integrating both sides, we get:
u = ∫ v dx
Substituting this into equation 1, we get:
4v'' + 12v' = 1
We find the solutions of the homogeneous equation:
4v'' + 12v' = 0
The characteristic equation of this equation is:
4r + 12 = 0
r = -3
Therefore, the general solution is:
v_h = C3 e^(-3x)
To find a particular solution, we assume a form of:
v_p = A e^(-3x) + B
Substituting this into the differential equation, we get:
4(-3A) e^(-3x) = 1
Therefore, A = -1/12.
Substituting this into v_p, we get:
v_p = -1/12 e^(-3x) + B
To find u_p, we integrate v_p with respect to x:
u_p = ∫ v_p dx = -1/36 e^(-3x) + Bx + C
Therefore, the particular solution is:
y_p = (-1/36 e^(-2x) + Bx + C) e^x cos(x) +C
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Related Questions
Which equation has these ordered pairs as solutions?
(1,7), (3,9), (5,11)
Answers
RS
ols
Two lines meet at a point that is also the endpoint of a ray as shown.
w
Jes
120°
is
What are the values of w, z,and y? What are some of the angle relationships? Select your answers from the drop-
down lists
35
The angles with measurements w' and 120 are vertical
The value of y is
The angle that measures a' is vertically opposite from the angle that measures
Thus, the value of wis ✓
degrees. Thus, the value of z
Answers
1. The angles w and 120 are supplementary angles
2. The value of w is 60 degrees
3. a is vertically opposite to angle 120
4. y is 25 degrees
What are vertically opposite angles?
Vertically opposite angles, also known as vertical angles, are a pair of angles formed by two intersecting lines. Vertical angles are opposite to each other and share a common vertex but not a common side.
1) 120 + w = 180 (Supplementary angles)
2)w = 60 degrees
3) a = 120 (Vertically opposite angles)
4) y = 180 - (120 + 35)
y = 25 degrees
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Find the slope of the line that passes through each pair of points (-3,4) and (2,-3)
Answers
Answer:
-7/5
Step-by-step explanation:
(-3 - 4) / (2 - -3) = -7/5
Given the quantities a=3.7m, b=3.7s, c=80m/s, what is the value of the quantity d=a^3/cb^2?
Answers
The value of the quantity d is 0.0462 m^2/s.
Here,
The quantities a=3.7m, b=3.7s, c=80m/s.
We have to find the value of d = a^3/cb^2
What is quantity?
A quantity is an amount, number, or measurement that answers the question 'how much?' Quantities can be expressed in numbers or non-standard units.
Now,
The quantities a=3.7m, b=3.7s, c=80m/s.
The value of d;
\(d = \frac{a^{3} }{cb^{2} }\)
\(d = \frac{3.7*3.7*3.7 }{80 * 3.7^{2} }\)
\(d = \frac{3.7}{80}\)
\(d = 0.0462\)
Hence, The value of the quantity d is 0.0462 m^2/s.
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Think About the Process You have 40 quarters. You find 20% more quarters in your room. Then
you go shopping and spend 75% of the total number of quarters. Write an expression to represent the
total number of quarters you take with you shopping. Calculate, in dollars, the amount of money you
have left.
What expression represents your total number of quarters?
OA. 40 + (40 +0.75)
OB. 40 + (40 +20)
OC. 40 + (40 x 20)
OD. 40+ (40 x 0.20)
Answers
Answer:A. 1.4*20 quarters
B.14
Step-by-step explanation:
you start with 20 quarters= 20*1
you find 40% of twenty quarters in your room= 4/100*20=0.4*20
20*1+20*0.4= 20*1.4
B
1.20*1.4=28
2.28*50/100=14
The solution is Option D.
The equation to express the total number of quarters is
A = 40 + ( 40 x 0.20 )
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation for total number of quarters be = A
Now , the value of A is calculated by
a)
The number of quarters initially = 40 quarters
The percentage of quarters found in room = 20 % of number of quarters
Percentage of quarters found in room = 20/100
So , The total number of quarters you have is =
number of quarters initially + percentage of quarters found in room x number of quarters initially
The equation will be
The total number of quarters you have is = 40 + ( 40 x 20/100 )
The total number of quarters you have is = 40 + ( 40 x 0.20 )
Therefore , the equation is A = 40 + ( 40 x 0.20 )
b)
Now , the number of quarters you have before going shopping is
A = 40 + ( 40 x 0.20 )
A = 40 + ( 8 )
A = 48 quarters
Now , there are 4 quarters in 1 dollar
So , 48 quarters is 12 dollars
Hence , The number of quarters you have before going shopping is 48 quarters or $ 12
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what is the solution?
Answers
Answer: x = 36
Step-by-step explanation:
\(\frac{2}{3} x-4=20\\\\\frac{2}{3} x=20+4\\\\\frac{2x}{3} =24\\\\2x=24*3\\\\x=\frac{72}{2} =36\)
which statement best describes the variance in a data set? it typically increases along with the mean. it is the size of the sample. it is another term for median. it is equal to the correlation coefficient. it is the square of the standard deviation.
Answers
The statement that best describes the variance in a data set is "it is the square of the standard deviation." Variance is a measure of how spread out the data is from the mean, and it is calculated by finding the average of the squared differences from the mean.
The standard deviation is the square root of the variance and represents the average distance from the mean. Therefore, the variance is the square of the standard deviation. The other statements are not accurate descriptions of variance. The variance is not related to the size of the sample or the correlation coefficient, and it does not increase along with the mean. The median is a measure of central tendency, not variability.
The statement that best describes the variance in a data set is: it is the square of the standard deviation. Variance measures the dispersion of data points from the mean, and it helps to understand the spread in the data set. It is not the size of the sample, nor another term for median, nor equal to the correlation coefficient. The variance does not typically increase along with the mean, as it is a separate measure of dispersion. Standard deviation is the square root of variance, making variance the square of the standard deviation.
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Y=4x-2 and y=-4+1 is.perpinducular or.not?
Answers
Answer:
No
Step-by-step explanation:
These are written in slope-intercept form:
\(y=mx+b\)
m is the slope and b is the y-intercept
When two lines are perpendicular, their slopes will be opposite reciprocals of each other. For example:
\(y=-2x+5\\\\y=\frac{1}{2}x-9\)
The lines above are perpendicular because the slopes:
\(m=-2\\\\m=\frac{1}{2}\)
are opposite reciprocals.
In the given lines, the slopes are not opposite reciprocals:
\(m=4\\\\m=-4\)
So they are NOT perpendicular lines.
:Done
need help with number 9 pls
Answers
Answer:
i) a = 2, b = - 3, c =-6
Step-by-step explanation:
4x² - 12x + 3
4x² - 6x - 6x + 9 - 6
(2x - 3)(2x - 3) - 6
(2x - 3)² - 6
(2x + (- 3) )² - 6
In form of (ax + b)² - 6
a = 2, b = - 3, c =-6
A house cost $320,000 in 2005. By the year 2019, its value was $560,000. What was the growth rate as a percent for that 14-year period? (Remember, i=(p^2/p^1)^1n−1
Answers
Answer:
Growth rate = 4.08%
Step-by-step explanation:
Rate of interest 'i' for the growth in the cost of the house,
i = \((\frac{\text{Final amount}}{\text{Initial amount}})^{\frac{1}{n}}-1\)
Here 'n' = Duration or time (in years)
i = \((\frac{560000}{320000})^{\frac{1}{14}}-1\)
i = \((\frac{7}{4})^{\frac{1}{14}}-1\)
i = 1.0408 - 1
i = 0.0408
i = 4.08%
Therefore, growth rate for this period is 4.08%.
Please help! On a field trip, there are 3 chaperones for every 20 students. There are 92 people on the trip.
Part a: How many chaperones are there?
Part b: how many students are there?
Answers
Answer:
A: 12 B: 80
Step-by-step explanation:
92 - 80 = 12
80 students
12 Chaperones
Answer:
12 chaperones and 80 students
Step-by-step explanation:
Start by multiplying the numbers seperatly and then add. Keep doing this till you get to 92.
20(2)=40. 3(2)=6 46
20(3)=60. 3(3)=9 69
20(4)=80. 3(4)=12 92
Now you know there are 12 chaperones and 80 students. There are other ways to answer this problem like using systems of equations but it still gives the correct answer and is less advanced.
Hope this helps:)
I need help badly there are more problems
Answers
The missing angle represented by 6 is 107 degrees
What is supplementary angleSupplementary angles are a pair of angles that add up to 180 degrees. In other words, if you have two angles that, when combined, create a straight line, then they are supplementary angles.
Angles on a straight line are supplementary and hence the potion that has 6 is equal to say x
x + 73 = 180
x = 180 - 73
x = 107 degrees
Supplementary angles can be found in many geometric shapes, such as triangles, quadrilaterals, and polygons.
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Convert the polar equation racos(20) = 10 to a rectangular equation in terms of x and y).
Answers
We can use the relationship between polar and rectangular coordinates. The rectangular coordinates (x, y) can be related to the polar coordinates (r, θ) through the equations x = rcos(θ) and y = r*sin(θ).
For the given equation rcos(θ) = 10, we can substitute x for rcos(θ) to obtain x = 10.
This means that the x-coordinate is always 10, regardless of the value of θ.
In summary, the rectangular equation in terms of x and y for the polar equation r*cos(θ) = 10 is x = 10, where the x-coordinate is constant at 10 and the y-coordinate can take any value.
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Find the sum of the interior angles of the shape. Use the remaining angles to solve for x. Polygons Help118105759130Sum of interior angles =degreesdegrees
Answers
Answer:
x = 112
Explanation:
The figure given is a pentagon; therefore, the sum of interior angles must add up to 540 degrees.
This means
\(105^o+118^o+75^o+130^o+x=540^o\)Adding the angles on the left-hand side gives
\(428^o+x=540^o\)Finally, subtracting 428 from both sides gives
\(428^o+x-428^o=540^o-428^o\)\(x=540^o-428^o\)\(x=112^o\)which is our answer!
Help anyone can help me do the question 4,I will mark brainlest.
Answers
Answer:
Hello,
Step-by-step explanation:
\(A=(1,2)\\B=(0,-1)\\\overrightarrow{AB}=((0,-1)-(1,2)=(-1,-3)\ ||\overrightarrow{AB}||^2=1+9=10\\\overrightarrow{BC}=((3,-2)-(0,-1)=(3,-1)\ ||\overrightarrow{BC}||^2=9+1=10\\\\Triangle\ is\ isosceles.\\\\\overrightarrow{AB}.\overrightarrow{BC}=(-1,-3)*\left[\begin{array}{c}3\\-1\end{array}\right] =-3+3=0\\\\Triangle \ is\ right.\\\\\)
Answer:
(Work below)
BRAINLIEST, PLEASE!
Step-by-step explanation:
A(1, 2) B(0, -1) C(3, -2)
The distance between A and B is:
3^2 + 1^2 = c^2
9 + 1 = c^2
c = 10^1/2
The distance between B and C is:
1^2 + 3^2 = c^2
1 + 9 = c^2
c = 10^1/2
The distance between C and A is:
4^2 + 2^2 = c^2
16 + 4 = c^2
c= 20^1/2
Since AB and BC are the same, but AC isn't, the triangle is isosceles.
What are all the like terms of 5x
Answers
Answer:
a number times 5
Step-by-step explanation:
Find the mean, median and standard deviation of the
following numbers. Which of the mean and median will give
the best average?
11, 12, 14, 16, 17, 17, 18
Answers
Answer:
the mean is 15
median is 16
best average would be mean since its all added together and divided by #
Step-by-step explanation:
to find mean add all numbers and divide by amount of numbers
median is middle number.
Answer all the questions Question One a. Show the equations for calculating 1. Bulk Volume of a reservoir in ft 3 and barrels 2 . Pore Volume of a reservoir in ft 3 and barrel 3 . Hydrocarbon Pore Volume in ft 3 and in barrel.
Answers
The equations for Bulk Volume of a reservoir in ft³ is VB = A*h and in barrels is VB = (A*h) / 5.615. The equations for Pore Volume of a reservoir in ft³ is VP = φ*VB and in barrels is VP = (φ*VB)/5.615. The equations for Hydrocarbon Pore Volume in ft³ is VHC = φ*S*VB and in barrels is VHC = (φ*S*VB)/5.615.
The equations for calculating the bulk volume, pore volume, and hydrocarbon pore volume of a reservoir are as follows:
1. Bulk Volume (VB):
In cubic feet (ft³):VB = A * hIn barrels (bbl):VB = (A * h) / 5.615Where:
VB = Bulk Volume
A = Cross-sectional area of the reservoir in square feet (ft²)
h = Thickness of the reservoir in feet (ft)
2. Pore Volume (VP):
In cubic feet (ft³):VP = φ * VBIn barrels (bbl):VP = (φ * VB) / 5.615Where:
VP = Pore Volume
φ = Porosity of the reservoir (dimensionless)
VB = Bulk Volume
3. Hydrocarbon Pore Volume (VHC):
In cubic feet (ft³):VHC = φ * S * VBIn barrels (bbl):VHC = (φ * S * VB) / 5.615Where:
VHC = Hydrocarbon Pore Volume
φ = Porosity of the reservoir (dimensionless)
S = Saturation of hydrocarbons in the reservoir (dimensionless)
VB = Bulk Volume
The conversion factor from cubic feet (ft³) to barrels (bbl) is 5.615.
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If 45% of a number , e is 180 , what is 95% of e
Answers
Answer:
380
Step-by-step explanation:
Find 1% of e:
180 / 45 = 4
Find 95% of e:
4 * 95 = 380
I don't know which one to pick.
Answers
Answer:
I think the answer Is k=4,y=4x
Find the area of the sector in
terms of pi.
12
210°
Area
=
[?]
T
![Find the area of the sector interms of pi.12210Area=[?]T](https://i5t5.c14.e2-1.dev/c-qa-images/contents/attachments/xO7GlIDtpvxPaQdPAPSb7bliiKE7aIcQ.png)
Answers
A= 210/360 x π(12)^2
Answer:
84pi
Step-by-step explanation:
Sector Area = (pi x r^2 x *angle*)/ 360
(pi x 12^2 x 210)/ 360
= 84pi
= 263.893
surface area of triangular prism 5 in 4 in 8 in 2 in
Answers
The Total surface of triangular prism is 112 inches.
Surface area calculation.
To calculate the surface area of a triangular prism, you need the measurements of the base and the height of the triangular faces, as well as the length of the prism.
The given measurements are;
Base ; 5 inches and 4 inches
height is 8 inches
Length of the prism is 2 inches.
To find the total surface area, we sum up the areas of all the faces:
Total surface area = area of triangular faces + area of rectangular faces + area of lateral faces.
area of triangular faces = 5 inches × 4 inches = 20 inches.
area of the two faces = 20 ×2 =40
Area rectangular faces = 5 inches × 8 inches/ 2 = 40 inches.
Area of lateral faces = 8 inches ×2 = 16 square inches
for the two lateral faces is 16 × 2 = 32 square inches.
Total surface area = 40 square inches + 40 inches + 32 square inches = 112 square inches.
The Total surface of triangular prism is 112 inches.
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I need help please, question in photo
Answers
Answer:
A. and F. and B.
:)
Lin's goal is to drink 8 cups of water every day she drank 37 ounces before lunch today How much more water does Lin need to drink today to reach her goal?
Answers
there is 8 ounces in a cup so 8 x 8 = 64. she needs to drink 64 ounces that day and she already consumed 37 ounces. 64-37=27
Answer: Lin needs to drink 27 ounces of water to reach her goal.
Step-by-step explanation: 1 cup = 8 fluid ounces.
8 cups = 8 x 8 = 64 ounces.
As of now Lin has drank 37 ounces of water. So, this means till now she has drank 37/8 = 4.625 cups of water.
In terms of ounces, she needs to drink 64-37=27 ounces of water to reach her goal.
Hence, the answer is 27 ounces or 8-4.625=3.375 cups.
12.7 = x – 3.8 help again
Answers
Answer:
x=16.5
Step-by-step explanation:
12.7 + 3.8 = x - 3.8 + 3.8
16.5 = x
Answer:
x = 16.5
Step-by-step explanation:
In this equation, we're trying to figure out the value of x, or what it's equivalent to in this particular situation. Let's look at what we have right now:
12.7 = x - 3.8
How can we get x by itself? Well, the thing that immediately comes to my mind is adding 3.8 to both sides. This will isolate x and help is figure out what it's equal to. So, we start out with:
12.7 = x - 3.8
...and then we add 3.8 to both sides to get:
16.5 = x
And that's the value of x. We can check our answer by replacing x with 16.5 and seeing if both sides of the equation are equivalent:
12.7 = 16.5 - 3.8?
12.7 = 12.7
So, we're correct. If you need any more help, let me know! :)
in the xy-plane, which of the following is an equation of a vertical asymptote to the graph Of y=sec(6x-pi)? (A) x=pi/6 (B) x=pi/4 (C) x=pi/3 (D)=x=pi/2 (E) x=pi
Answers
The equation of a vertical asymptote to the graph of y = sec(6x - π) is x = π/6. Hence, option a is correct.
The function y = sec(6x - π) has vertical asymptotes at the values of x where the denominator of sec(6x - π) becomes zero. The reciprocal of sec(θ) is cos(θ). Because the cosine function has the values π/2, 3π/2, 5π/2, we will insert such an input that we get 0 in denominator.
6x - π = π/2
Solving for x,
6x = π/2 + π
6x = 3π/2
x = (3π/2) / 6
x = π/6
Therefore, the equation of a vertical asymptote to the graph of y = sec(6x - π) is x = π/6.
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Someone help please❤️ & Thankyou!!
Answers
Answers:
300 = 12i+36si = 10Yes it's reasonablei = -11Refer to the diagram below.
The explanation for each part is below as well.
========================================================
Part 1
i = number of individual nail polish bottles
s = number of sets of 4
12i = money earned from just selling the individual bottles only
36s = money earned from just selling the sets of 4 only
12i+36s = total money earned = 300
12i+36s = 300 is the equation to set up
This is the same as 300 = 12i+36s because A = B is the same as B = A.
========================================================
Part 2
Plug in s = 5 to represent she sold 5 sets. Let's solve for i
12i+36s = 300
12i+36(5) = 300
12i+180 = 300
12i = 300-180
12i = 120
i = 120/12
i = 10
If she wants to sell $300 worth of product overall, and she already sold 5 sets so far, then she needs to sell 10 individual bottles to reach this monthly goal.
========================================================
Part 3
Yes it's reasonable to reach this goal. This is because the value of i (from part 2) was some positive whole number.
========================================================
Part 4
We'll repeat part 2 but with s = 12
12i+36s = 300
12i+36(12) = 300
12i+432 = 300
12i = 300-432
12i = -132
i = -132/12
i = -11
As the instructions state, we get a negative value. This is not reasonable because we can't sell a negative number of items. A negative number of items in general doesn't make sense.
The instructions further state that $432 was already made and that she exceeded her goal. Meaning that she doesn't have to worry about selling the individual bottles if selling the sets has her reach something over $300.
A red kangaroo hops at an average speed of 20km per hour. How fast is this in meters per minute?
Answers
Answer:
333.3 meters per minute
Step-by-step explanation:
The best way to solve this problem is using dimensional anaysis. First, we write out our starting units, that being 20km/1hr. We have to keep in mind that we want to change the kilometers to meters and the hours to minutes.
\(\frac{20km}{1hr}\)
We know that there are 1000 meters in 1 kilometer. We add this to the dimensional analysis as 1000m/1km. We write it as this because we want the kilometers to cancel each other out. We only want the meters.
\(= \frac{20km}{1hr} *\frac{1000m}{1km} ...\)
We also know that 1 hour is 60 minutes. We add this to the analysis as well so that the hours cancel each other.
\(= \frac{20km}{1hr} *\frac{1000m}{1km} * \frac{1hr}{60min}\)
We now solve this expression. Since both the kilometers and the hours cancel out, we have meters per minute as our unit. All that's left are the numbers.
= (20*1000*1)/(1*1*60) m/min
= 333.3 meters per minute
Pls help me this was due a long time ago ❤️
Answers
Answer:
its the last one
Step-by-step explanation:
Answer:
2017 - 2019
Step-by-step explanation:
because 2019 is the largest and 2017 is the largest one from the top of the weight list
I dont know is this right
What is 2x^2=18 Solve it please!
Answers
Answer:
3
Step-by-step explanation:
\(2x^{2} = 18\\ => x^{2} = \frac{18}{2} =9\\ => x = \sqrt{9} = 3\)
Hope u'll like it.....
Step-by-step explanation:
2x²=18
x²=18/2
x²=9
squaring both sides
and we get
x=3