The Second Linearly Independent solution of the differential equation is:
y = c₁x⁷ + c₂(Ax¹⁰ + Bx⁻¹¹)
When solving a second-order linear differential equation of the form
x²y'' - 42y = 0, it is important to find two linearly independent solutions to fully describe the general solution. The first solution is given as y₁=x⁷.
To find the second linearly independent solution, we can use the method of reduction of order.
Let y₂ = u(x)y₁(x), where u(x) is a function to be determined.
Then we have y₂' = u(x)y₁'(x) + u'(x)y₁(x) and y₂'' = u(x)y₁''(x) + 2u'(x)y₁'(x) + u''(x)y₁(x).
Substituting y₂ and its derivatives into the original differential equation, we have:
x²(u(x)y₁''(x) + 2u'(x)y₁'(x) + u''(x)y₁(x)) - 42u(x)y₁(x) = 0
Dividing by x²y₁(x), we get:
u''(x)/u(x) + 2/x[u'(x)/u(x)] - 42/x² = 0
Let v(x) = u'(x)/u(x), then v'(x) = u''(x)/u(x) - (u'(x))²/(u(x))². Substituting v(x) into the above equation, we have:
v'(x) + 2/xv(x) - 42/x² = 0
This is now a first-order linear differential equation that can be solved using an integrating factor. Letting mu(x) = x², we have:
(x²v(x))' = 42
Solving for v(x), we get:
v(x) = 21/x + C/x²
where C is an arbitrary constant. Substituting back to u(x), we get:
u(x) = Ax³ + Bx⁻⁻¹⁸
where A and B are constants. Therefore, the second linearly independent solution is
y₂ = (Ax³ + Bx⁻¹⁸)x⁷ = Ax¹⁰ + Bx⁻¹¹
Hence, the general solution of the differential equation is:
y = c₁x⁷ + c₂(Ax¹⁰ + Bx⁻¹¹)
where c₁ and c₂ are arbitrary constants
To know more about differentiation here
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what is 3333 x 33 x 23 - 908
Answer:
2528839
Step-by-step explanation:
your welcome
The 2010 U.S. Census asked every household to report information on each person living there. Suppose for a sample of 30 households selected, the number of persons living in each was reported as follows. 2 3 1 2 6 4 2 1 5 3 2 3 1 2 2 1 3 1 2 2 4 2 1 2 9 3 2 1 1 3 Compute the mean, median, mode, range, lower and upper quartiles, and interquartile range for these data. (Round the mean to 2 decimal places.) Mean
Answer:
[tex]\bar x =2.53[/tex] -- Mean
[tex]Median= 2[/tex]
[tex]Mode = 2[/tex]
[tex]Range = 8[/tex]
[tex]Q_1 = 1[/tex]
[tex]Q_3 = 3[/tex]
[tex]IQR = 2[/tex]
Step-by-step explanation:
Given
n = 30
Data: 2 3 1 2 6 4 2 1 5 3 2 3 1 2 2 1 3 1 2 2 4 2 1 2 9 3 2 1 1 3
Solving (a): The mean
This is calculated as:
[tex]\bar x =\frac{\sum x}{n}[/tex]
[tex]\bar x =\frac{2 +3 +1 +2 +6 +4 +2 +1 +5 +3 +2 +3+ 1+ 2 +2 +1 +3 +1 +2 +2 +4 +2 +1 +2 +9 +3 +2 +1 +1 +3}{30}[/tex]
[tex]\bar x =\frac{76}{30}[/tex]
[tex]\bar x =2.53[/tex]
Solving (b): The median
First arrange the given data
Arranged: 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 5 6 9
[tex]Median =\frac{1}{2}(n+1)[/tex]
[tex]Median =\frac{1}{2}(30+1)[/tex]
[tex]Median =\frac{1}{2}(31)[/tex]
[tex]Median = 15.5th[/tex]
This implies that the median is the average of the 15th and 16th item
[tex]Median = \frac{2+2}{2}[/tex]
[tex]Median = \frac{4}{2}[/tex]
[tex]Median= 2[/tex]
Solving (c): The mode
From the given data; 2 has the highest frequency of 11.
So,
[tex]Mode = 2[/tex]
Solving (d): The Range
[tex]Range = Highest\ Data - Lowest\ Data[/tex]
The highest is 9 and the lowest is 1,
So:
[tex]Range = 9 - 1[/tex]
[tex]Range = 8[/tex]
Solving (e): Lower (Q1) and Upper (Q3) quartile.
Q1 is calculated as:
[tex]Q_1 = \frac{1}{4}(n+1)th[/tex]
[tex]Q_1 = \frac{1}{4}(30+1)th[/tex]
[tex]Q_1 = \frac{1}{4}(31)th[/tex]
[tex]Q_1 = 7.75th[/tex]
[tex]Q_1 = 8th\ item[/tex]
[tex]Q_1 = 1[/tex] -- from the arranged data
Q3 is calculated as:
[tex]Q_3 = \frac{3}{4}(n+1)th[/tex]
[tex]Q_3 = \frac{3}{4}(30+1)th[/tex]
[tex]Q_3 = \frac{3}{4}(31)th[/tex]
[tex]Q_3 = 23.25th[/tex]
[tex]Q_3 = 23rd\ item[/tex]
[tex]Q_3 = 3[/tex] -- from the arranged data
Solving (f): The interquartile range (IQR)
[tex]IQR = Q_3 - Q_1[/tex]
[tex]IQR = 3-1[/tex]
[tex]IQR = 2[/tex]
Please help, this is due soon.
если x=6 то y=
если y=-4 то x=
Step-by-step explanation:
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Answer:
X=6 To y=-6
y=-4 TO X = 4
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What are the next three terms in the sequence?
-6,5,16,27...
38, 49,60
37,47,57
36, 45, 54
36, 46,57
Answer:
uff si está bien fácil por favor tu te sabes la respuesta y yo tambien
The ratio of girls to boys on a soccer team is 2 to 3. If there are 15 boys on the team, how many are girls? Let g = the number of girls on the team. Set up a proportion for this word problem.
Answer:
3X=15
X=5
2X5= 10
G=10
Step-by-step explanation:
1 over blank = 2 over 10 = blank over 100
Answer:
[tex]\frac{1}{5}[/tex] = [tex]\frac{2}{10}[/tex] = [tex]\frac{20}{100}[/tex]
Step-by-step explanation:
It's all about equivalent fractions.
[tex]\frac{1}{5}[/tex]
[tex]\frac{2}{10}[/tex]
[tex]\frac{20}{100}[/tex]
[tex]\frac{1}{5}[/tex] 1 × 2 = 2. 5 × 2 = 10. That is what makes [tex]\frac{2}{10}[/tex].
[tex]\frac{2}{10}[/tex] 2 × 10 = 20. 10 × 10 = 100. That is what makes [tex]\frac{20}{100}[/tex].
Pls help due in 10 minutes
Answer:
Boy I'm really boutta get to yo pickle chin ahh boy, egg head like collard greens head a//ss boy, oh hell nah boy yo dirt ahh boy stank ahh boy afro head ahh, lip gloss chin ahh boy ugly ahh boi
Step-by-step explanation:
Aawdadakdaw da awdwa dadeefdcdcs
Answer plzzzzzzzzzzzzz
Answer:
x=12.727822
Step-by-step explanation:
Since we know two of the angles, we can find the third. 180-90-45=45. This means the two sides of the triangle are also equal, and we know the hypotenuse is 18. So we can use the Pythagorean theorem.
x²+x²=18²
2x²=324
x²=162
x=√162
x=12.727822
help with geometry please !
Ms. Gonzalez buys folders for her office. She buys 4 small folders for every 1 large
folder. She buys 15 folders in all. Small folders cost $0.75, and large folders cost
$1.25. How much more money does Ms. Gonzalez spend on small folders than
large folders? Show your work.
Answer:
Step-by-step explanation:
The amount of money that Ms. Gonzalez spends on small folders than large folders will be $12.75.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
Ms. Gonzalez buys folders for her office. She buys 4 small folders for every 1 large folder. She buys 15 folders in all. Small folders cost $0.75, and large folders cost $1.25.
Let 'x' be the number of small folders and 'y' be the large folders. Then the equations are given as,
x + y = 15 ...1
y = 4x ...2
From equations 1 and 2, then we have
x + 4x = 15
5x = 15
x = 3
Then the value of 'y' is given as,
3 + y = 15
y = 12
The cost of the small folder is given as,
⇒ $0.75 × 3
⇒ $2.25
The cost of the large folder is given as,
⇒ $1.25 × 12
⇒ $15
The difference is given as,
⇒ $15 - $2.25
⇒ $12.75
The amount of money that Ms. Gonzalez spends on small folders than large folders will be $12.75.
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PLEASE HELP ME WITH 9 (and 7a./7b. If you can) :)
Answer:
y = (1/4) 2^x
Step-by-step explanation:
Remark
y = a * x^n is the general equation. Let's see what that means.
Solution
Solve for x
t_12 = a x^12
1024 = a x^12
t_8 = a x^8
64 = a x^8
Now divide the two equations.
1024/64 = a x^12/a x^8 Cancel out the as
1024 / 64 = x^12 / x^8
16 = x^(12 - 8)
16 = x^4 Take the 4th root of both sides.
[tex]\sqrt[4]{x^4}[/tex] = [tex]\sqrt[4]{16}[/tex]
x = 2
Go back and solve for a
y_12 = a x^12
1024 = a 2^12
1024 = a 4096 Divide by 4096
1024 / 4096 = a
a = 1/4
Problem 7
a) First of negative numbers are probably not a good idea. It means that the population is going to decrease. That's not what will happen if you take care of the fruit flies.
So for the domain, 0 is a good starting point.
That will give you 100 flies on day 0. That's how many you bought.
I'm not a biologist so I have no idea in the world how long a fruit fly will live even with the best of care. I'm going to try 60 days and see what happens.
y = 100 * 1.05 ^60
y = 100 * 18.68
y = 1868
Is this number reasonable? Even with the biggest olive jar you can find, and the best care, I don't think that number is realistic. It's too large. Eventually all will die of disease because they are living in their own waste. So Let's try 20 days
y = 100 * 1.05^20
y = 100 * 2.65
y = 265
That's a lot more reasonable. So for convenience sake let us say that 265 flies could live for 20 days
So to answer the question
The domain would be 0 to 20
The range would be 100 to 265
b)
2 weeks = 2*7 days
2 weeks = 14 days
y = 100 * 1.05^14
y = 100 * 1.98
y = 198 fruit flies.
Evaluate (–6)^2 – 7^2.
A) 85
B) 169
C) –85
D) –13
The function h is defined by h(x)=x^2 +2.
Find h(3y).
To test the effectiveness of a drug proposed to relieve symptoms of depression, psychiatrists recruited volunteers for clinical tests. All of the volunteers were diagnosed with symptoms of depression. The psychiatrists provided the drug to first group and a sugar pill to the second group. In the description of the above situation, determine the control group.
Answer:
The control group is the group that takes the sugar pill
Step-by-step explanation:
In drug testing, to determine the efficacy of a drug, different groups of experimental subjects are recruited, one group is given the drug to be tested (treatment group) while another group is given a placebo.
A placebo is a fake version of the drug to be tested that does not have any therapeutic effect on the test subjects that receive it. In this case, the control group receives the placebo which is the sugar pill. They serve as the baseline of measurement when the outcome of the effect of the drug testing is done. The groups that received the actual drugs are compared against the groups that received the hoax treatment, and any change occuring in the test subjects become the changes occuring as a result of the test drug used, when the observations on the control group has been ruled out.
Two cards are selected from a standard deck of 52 playing cards. The first card is not replaced before the second card is selected. Find the probability of selecting a jack and then selecting a king
The probability of selecting a jack and then selecting a king is _______
(Round to three decimal places as needed)
Answer:
3.25
Step-by-step explanation:
Rewrite the expression by factoring out (w-2).
5w(w – 2) + 2(w-2)
Find an equation of a line parallel to the line y = 1/2x-3that contains the point (6,4).
Write the equation in slope-intercept form.
#First find the gradient first, remember that the formula of equation of the line is y = mx + c and gradient's symbol is m
y = ½x - 3
#so the gradient is ½, now u can find the equation of the lime.
y - y1 = m(x - x1)
y - 4 = ½(x - 6)
y - 4 = ½x - 3
y = ½x - 3 + 4
y = ½x + 1
Answer : y = ½x + 1Plsss help me!!!Mei graphed the locations of several places in her town on the coordinate plane shown below. There is also
an ice cream shop halfway between the school and hospital.
At what coordinates should Mei graph the ice cream shop?
Answer:
(6, -4)? is this on khan academy or smth
Step-by-step explanation:
solve for x 0°<x<360
3cos2x +sinx=1
Step-by-step explanation:
3cos2x +sinx=1
3(1-2sin²x)+sinx=1
3-6sin²x+sinx=1
sinx(-6sinx+1)=1
either
sinx=1
sinx=sin(90)
:.x=90
or
-6sinx+1=1
-6sinx=1-1
sinx=0/-6
sinx=sin0,sin180
:.x=0,180and90
Which number is prime?
A. 4
B. 10
C. 22
D. 31
Answer:
d. 31
Step-by-step explanation:
BRAINLIEST PLEASE
100 POINTS!!
Find the slope and the y intercept of the line in the graph.
Answer:
Slope: 5/1
y-intercept: 2
Step-by-step explanation:
write the linear function. i will make brainylist
Answer:
y=-2x+5
Step-by-step explanation:
To figure out the equation of a line, use the formula y=mx+c, where y and x represent the co-ordinate values, m is the slope and c is the y-intercept (where the graph cuts the y-axis).
The slope can be found by taking two points (x1,y1) and (x2,y2), and using the formula m=(y2-y1)/(x2-x1):
(0,5) and (1,3)
m=(3-5)/(1-0)
m=-2
We can see from the graph that the line cuts the y-axis at y=5, so c=5.
Now that we know m and c, we plug these values into our formula to get y=-2x+5
last year, Nicole biked n miles. This year, she biked 524 miles. Using n, write an expression for the total number of miles she biked.
Answer:
n+524
Step-by-step explanation:
you add both years worth of biking together!
X < -4.8
Chose 2 Answers:
(A)2
(B)0.9
(C)-2.3
(D)-5
(E)-6.1
Please Help.
Answer:
D & E
Step-by-step explanation:
Penny is the manager of a nursery for pre-school children.
She knows that the ratio of the number of adults to the number of children must be 1 : 8
Penny has 56 children at the nursery.
Work out the number of adults Penny needs for 56 children?
Show your wor
Answer:
9
Step-by-step explanation:
If a sprinkler waters 1/20 of a lawn in 1/5 of an hour how much time will it take to water the entire lawn?
• 3/20 hour
• 1/4 hour
• 4 hours
• 5 hours
Answer:
4 hours
Step-by-step explanation:
Can you please help thanks
Answer:5/2a + 10/3b
Step-by-step explanation:
(1/6)(15a+20b)
=(1/6)(15a)+(1/6)(20b)
write the digit on the blank which occupies the place value
Answer:
guganitckihkgucifufu
Will marks as brain Uh I don’t have to watch ads for like 40 questions and I don’t know if you just get them for answering questions or if I accidentally bought premium help