Use reduction of order. Given a solution [tex]y_1(x) = x^2[/tex], look for a second solution of the form [tex]y_2(x) = y_1(x)v(x)[/tex].
Compute the first two derivatives of [tex]y_2(x)[/tex]:
[tex]y_2 = x^2v \\\\ {y_2}' = x^2v' + 2xv \\\\ {y_2}'' = x^2v''+4xv' + 2v[/tex]
Substitute them into the ODE:
[tex]x^4 (x^2v'' + 4xv' + 2v) + x^3 (x^2v' + 2xv) - 4x^2 (x^2v) = 1 \\\\ x^6v'' + 5x^5v' = 1[/tex]
Now substitute [tex]w(x) = v'(x)[/tex] and you end up with a linear ODE:
[tex]x^6w'+5x^5w=1[/tex]
Multiply through both sides by [tex]\frac1x[/tex] (if you're familiar with the integrating factor method, this is it):
[tex]x^5w'+5x^4w = \dfrac1x[/tex]
Bear in mind that in order to do this, we require [tex]x\neq0[/tex]. Just to avoid having to deal with absolute values later, let's further assume [tex]x>0[/tex].
Notice that the left side is the derivative of a product,
[tex]\left(x^5w\right)' = \dfrac1x[/tex]
Integrate both sides with respect to [tex]x[/tex] :
[tex]x^5w = \displaystyle \int\frac{\mathrm dx}x \\\\ x^5w = \ln(x) + C_1[/tex]
Solve for [tex]w(x)[/tex] :
[tex]w = \dfrac{\ln(x)+C_1}{x^5}[/tex]
Solve for [tex]v(x)[/tex] by integrating both sides:
[tex]v = \displaystyle \int \frac{\ln(x)+C_1}{x^5} \,\mathrm dx[/tex]
Integrate by parts:
[tex]\displaystyle f = \ln(x) + C_1 \implies \mathrm df = \frac{\mathrm dx}x \\\\ \mathrm dg = \frac{\mathrm dx}{x^5} \implies g = -\frac1{4x^4} \\\\ \implies v = -\frac{\ln(x)+C_1}{4x^4} + \frac14 \int \frac{\mathrm dx}{x^5} \\\\ v = -\frac{\ln(x)+C_1}{4x^4} - \frac1{16x^4} + C_2 \\\\ v = -\frac{4\ln(x)+C_1}{16x^4}+C_2[/tex]
Solve for [tex]y_2(x)[/tex] :
[tex]\displaystyle \frac{y_2}{x^2} = -\frac{4\ln(x)+C_1}{16x^4}+C_2 \\\\ y_2 = -\frac{4\ln(x)+C_1}{16x^2} + C_2x^2[/tex]
But since [tex]y_1(x)=x^2[/tex] is already accounted for, the second solution is just
[tex]\displaystyle y_2 = -\frac{4\ln(x)+C_1}{16x^2}[/tex]
Still, the general solution would be
[tex]\displaystyle \boxed{y(x) = -\frac{4\ln(x)+C_1}{16x^2} + C_2x^2}[/tex]
2.53 x 0.0041
(please help)
Answer:
0.010373
Step-by-step explanation:
hope it helps!
Simplify the complex question: 5/8 / 4/5
Answer:
The choose 25/32
Step-by-step explanation:
[tex] \frac{ \frac{5}{8} }{ \frac{4}{5} } = \frac{5}{8} \times \frac{5}{4} = \frac{25}{32} [/tex]
I hope I helped you^_^
Answer:
A. [tex] \frac{25}{32} [/tex] (first option)
Step-by-step explanation:
[tex] \frac{ \frac{5}{8} }{ \frac{4}{5} } = \frac{5}{8} \div \frac{4}{5} [/tex]
[tex] \frac{5}{8} \times \frac{5}{4} = \frac{5 \times 5}{8 \times 4} [/tex]
[tex] = \frac{25}{32} [/tex] ✅
In a group 40% like football ,70% like cricket and 30% like both games
Step-by-step explanation:
Ans is 20 .
F union C whole complement is 20
The double number line shows the approximate number of ounces in a certain number of pounds:
Based on the number line, about how many ounces are there in 6 pounds?
24 ounces
80 ounces
96 ounces
112 ounces
Answer:
the answer is 96 ounces i think
The job paid $25 for every 2 hours of work. Write an equation that represents how much the jobs pays, y, for x hours of work.
Answer:
$12,5xy
Step-by-step explanation:
y = how much the jobs pays
x = hours of work
[tex]25 \div 2[/tex]
[tex] = 12.5xy[/tex]
let f(x)=2x-1 and g(x)=1/x. Find the value of f(g(7))
Answer:
-5/7
Step-by-step explanation:
We need to find g(7) first
All we have to is plug 7 in the place of x
g(7) = 1/7
Now we have f(1/7) = 2x - 1
f(1/7) = 2(1/7) - 1
f(1/7) = 2/7 - 1
Don't forget about common denominators
2/7 - 7/7 = -5/7
PMDAS
1. 24 + 8 + (9 x2) x 4 + 7
Answer:
[tex]\huge\boxed{\sf 111}[/tex]
Step-by-step explanation:
= 24 + 8 + (9 x 2) x 4 + 7
Parenthesis
= 24 + 8 + 18 x 4 + 7
Multiplication
= 24 + 8 + 72 + 7
Addition
= 32 + 72 + 7
= 104 + 7
= 111
[tex]\rule[225]{225}{2}[/tex]
Hope this helped!
~AH1807Peace!14 - 2(x + 8) = 5x - (3x - 34); Prove: x = -9
Step-by-step explanation:
14 - 2(x + 8) = 5x - (3x - 34)
14 -2x -16 = 5x -3x+34
-2x -2 = 2x+34
-2x-2x = 34+2
-4x = 36
x = 36/(-4)
x = -9
Howard pays $327 for one dozen collector’s edition baseball cards. About how much does he pay for each baseball card?
Answer:
Howard spent $27.25 for each baseball card.
Step-by-step explanation:
1 dozen = 12
Set up an equation:
327 ÷ 12
27.25
Answer:
27.25
Step-by-step explanation:
If I'm 4years and my brother is half of my age now I'm 44 how old is he ?
Answer:
42 years
Step-by-step explanation:
if u were 4 years then your brother will be 2 years
if u r 44 then u can see that the difference between your previous age aand your brothers was 2 years
so here u just need to substract 2
44- 2 that will be 42
100th term of 1, 4, 16
Answer: 64
Step-by-step explanation:
Please help me to solve this question
Answer:
(p+5)(p-2)
Step-by-step explanation:
We are looking for two numbers that multiply to -10 (the rightmost number) and sum to 3 (the middle number)
These are 5 and -2
So we write
(p ____)(p ____)
and fill in the blanks
(p+5)(p-2)
Check by FOILing:
p^2 -2p + 5p -10
And combine the two middle terms.
p^2 + 3p - 10
m={y:y is a multiple of 3,5<y<10}
Answer:
6,9
Step-by-step explanation:
The two multiples of 3
which are more than 5 and less than 10 are
6 and 9
Help!!!!will have brainley I need help pls!
Option A " how the temperature of a cup of coffee changes as time passes ".
Hope it helps you.
I'LL GIVE BRAINLIEST;
You deposit $400 each month into an account earning 5% annual interest compounded monthly.
a) How much will you have in the account in 20 years?
b) How much total money will you put into the account?
c) How much total interest will you earn?
Answer:
amount in 30 yrs is: $1787.0977
interest is for 20 yrs: $1387.0977
Step-by-step explanation:
Given data
principal amount = $400
rate = 5 % = 0.05
time period = 20 years
(Sorry if I'm wrong :( )
The diameter of a quarter is 0.955 inches. If you laid 1,000,000 quarters side to
side, how far would that be?
Mrs Martin has $7000 in her saving account. Alonzo has 1/10 as much money in his account as Mrs Martin. How much money does Alonzo have in his account?
Answer:
7000*1/10
In a calculator type in 7000*(1/10)
700
Step-by-step explanation:
What is p (salamander)?
Answer:
the image/question isnt there
Step-by-step explanation:
Answer:
Step-by-step explanation:
Can anyone plzz tell me the reasons for this false ones
Plzz help
Answer:
It is correct
Step-by-step explanation:
Has two solutions x = a and x = -a because both numbers are at the distance a from 0. You begin by making it into two separate equations and then solving them separately. An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.
What number is two less then nine hundred and one
Answer:
eight hundred and ninety nine
Answer:
eight hundred and ninety nine
Step-by-step explanation:
Find the expression for f(x) that makes the following equation true for all values of x.
(81^x)/9^5x-8=9^f(x)
Answer:
Step-by-step explanation:
As it is bad written, i suppose it is
[tex]\dfrac{81^x}{9^{5x-8}} =9^{f(x)}\\\\9^{2x-(5x-8)}=9^{f(x)}\\\\2x-5x+8=f(x)\\\\\\\boxed{f(x)= -3x+8}\\[/tex]
5 men can paint 2 identical houses in 3 days. Assuming
that all the men work at the same rate, how long
will it take 10 men to paint 8 such houses?
Plz help I’ll mark brainliest
Answer:
6 days
Step-by-step explanation:
If you like my answer than please mark me brainliest thanks
Answer:6 days
Step-by-step explanation:
because it means 5 men painted 1 house in 1.5days(3:2=1,5), so for 5 men to paint 8 houses they will need 12 days (1.5x8=12) divide 12 in half because there will be twice as much men(10) and you get 6 days
please help A data set has its first and third quartiles as 9 and 17, respectively.
Which of the following data points would be considered an outlier for the data set?
27
17
3
41
Answer: 27
Step-by-step explanation: hihihihihiiiiihihi
27 is considered an outlier for the data set.
What is quartiles?In statistics, a quartile exists as a variety of quantiles which divides the number of data points into four parts, or quarters, of more or less equal size. The data must be ordered from smallest to largest to calculate quartiles; as such, quartiles exist a form of order statistic.
A quartile separates data into three points—a lower quartile, median, and upper quartile—to form four sets of the dataset.
Hence, 27 is considered an outlier for the data set.
To learn more about quartiles refer to:
https://brainly.com/question/9260741
#SPJ2
Examine the tile pattern at right
b. The pattern grow by adding 1 tile above the tile and adding 1 tile at the right of the tile.
c. In figure 0, there will be 1 tile. We know this because in each successive figures a tile is added at the above and a tile is added to the right, so ineach preceeding figure the same is reduced. In figure 1, there are e tiles, so in figure 0, there will be 3-2 = 1 tile.
Is the relation given by the set of ordered pairs below a function? (Yes or No)
(0,2),(8,-2),(-4,6)(-3,2)(4,-3)
Find the inverse of f(x) = -4+7/2x
Answer:
2/7x + 8/7
Step-by-step explanation:
f(x) = -4 + 7/2x
y = -4+7/2x
Exchange x and y
x = -4 +7/2y
Solve for y
Add 4 to each side
x+4 = -4+4 +7/2y
x+4 = 7/2y
Multiply each side by 2/7
2/7(x+4) = 2/7 * 7/2y
2/7(x+4) = y
2/7x + 8/7
Which fraction is equivalent to -7/8
1. 8/-7
2. -7/-8
3. 8/7
4. 7/-8
Answer:
Your answer is answer no 4 = 7/-8
Answer:
4.
Step-by-step explanation:
7/-8
-7/8 is between -1 and 0.
So, we can eliminate 1 and 3.
We can also get rid of 2 because -7/-8 = 7/8, which is positive.
We are left with 4.
18. Solve x^2+ 6x - 16 = 0 by completing the square.
Please give detailed steps!!
Answer:
x1 = -8, x2= 2
Step-by-step explanation:
1. Write 6x as a difference
= x^2 + 8x - 2x - 16 = 0
2. Factor the expressions
= x ( x + 8 ) - 2x - 16 = 0
= x ( x + 8 ) - 2 ( x + 8 ) = 0
Factor out x + 8 from the expression
= ( x + 8 ) ( x - 2 ) = 0
3. When the product of factors = 0, at least one factor is 0
x + 8 = 0
x - 2 = 0
4. Solve the equation for x
x = - 8
x = 2
Therefore, the equation has 2 solutions :
x1 = -8, x2 = 2
A spelunkers starts his journey at -500 feet and ends up at -100 feet.What is his change in elevation?
Answer:
+400
Step-by-step explanation:
Round 8.32 to the nearest whole number.
Answer:
1_+6''?(oytvbjkufx,hjf7(-&&&