The homogeneous equation
y'' + y = 0
has characteristic equation
r² + 1 = 0
with roots at r = ± √(-1) = ± i, and so the characteristic solution is
y = C₁ cos(x) + C₂ sin(x)
Using the method of undetermined coefficients, suppose
y = a x cos(x) + b x sin(x)
(we would normally consider y = a cos(x) + b sin(x), but cos(x) and sin(x) are already part of the characteristic solution)
The derivatives are
y' = a cos(x) - a x sin(x) + b sin(x) + b x cos(x)
y' = (b - a x) sin(x) + (a + b x) cos(x)
and
y'' = -a sin(x) + (b - a x) cos(x) + b cos(x) - (a + b x) sin(x)
y'' = (-2a - b x) sin(x) + (2b - a x) cos(x)
Then
y'' + y = (-2a - b x) sin(x) + (2b - a x) cos(x) + a x cos(x) + b x sin(x)
7 sin(x) = -2a sin(x) + 2b cos(x)
and it follows that a = -7/2 and b = 0.
So, the general solution to the DE is
y = C₁ cos(x) + C₂ sin(x) - 7/2 x cos(x)
which means the only valid solutions from the given choices are
y = 7 sin(x)
y = 7 cos(x)
Evaluate the expression 5xy+ -2xy+y^2 when x=4 and y=-5
Answer:
Hope it will help you...
During a 12-hour car ride, each of 4 family members gets to choose the music for an equal amount of time. Write an equation, where t is time, to find out how long each member of the family is in charge of selecting the music.
Write an explicit formula for an, the nth term of the sequence 35,42,49
Hi!
We can see the numbers are increasing by 7, therefore your equation would be:
n = p + 7
"p" represents the previous number, and "n" represents the number you're trying to find.
Hope this helps! :D
If (k+1), 3k, (4k+2) are three consecutive terms of an AP then k = ?
The three consecutive terms in an AP are (k+1) ,3k and (4k+2)
We know that
If a,b,c are the three consecutive terms in an AP then b = (a+c)/2
now ,
Let a = k+1 , b = 3k and c = 4k+2
⇛3k = (k+1+4k+2)/2
⇛3k = (5k+3)/2
⇛2×3k = 5k+3
⇛6k = 5k+3
⇛ 6k-5k = 3
⇛ k = 3
The value of k = 3
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The table below shows the cumulative distance and the number of hours Ryan drove on vacation.
Hours Driving Distance Traveled (miles)
0 0
3 186
4 248
7 434
10 620
Based on the information in the table, at what speed did he drive (miles per hour)?
A.
69 miles per hour
B.
62 miles per hour
C.
52 miles per hour
D.
42 miles per hour
Answer: 62 miles per hour
Step-by-step explanation:
620/10 = 62
The speed of Ryan is given by the slope m = 62 miles per hour
What is an Equation of a line?The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the equation of line be represented as A
Now , the value of A is
Let the first point be P ( 3 , 186 )
Let the second point be Q ( 4 , 248 )
Now , the slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Substituting the values in the equation , we get
Slope m = ( 248 - 186 ) / ( 4 - 3 )
m = 62 miles per hour
Hence , the rate of change of speed is m = 62 miles per hour
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165 of the city library members are children if there are 750 total members
Answer:
Step-by-step explanation:
Not sure what you require but (165/750) * 100
= 0.22 * 100
= 22% are children
Solve for X : 3/8 = 27/x
Answer:
x = 72
Step-by-step explanation:
3/8=27/x
cross multiply
3*x=3x 27*8=216
divide
216/3=x
x=72
For an experiment, the temperature of a liquid must stay above 60.5°F. The starting temperature of the liquid is 88.6°F. It is placed in a cooler, which lowers its temperature 2°F each minute.
How many minutes can the liquid can stay in the cooler?
Select from the drop-down menus to correctly complete the statement. at least 14.05 /more than 14.05/ fewer than 14.05 /at most
14.05 at most
If it is saying at least, than it would be in the cooler for longer than it should be and drop below 60.5
If saying more than 14.05, it is basically the same thing where it is inside the cooler for longer than it should be
If saying at most, that means that 14.05 is the max it would be in the cooler and keeps the liquid above 60.5
Answer: im late but here you go
12/2/22
Step-by-step explanation:
Which of the following equations has no solution?
3x + 1 = 3x + 1
3x+1= 3x + 2
3x+1= 2x + 1
Answer:
3x+1=3x+1
0x=0
no solution
3x+1=3x+2
0×=1
no solution
3x+1=2x+1
x=0
X IS THE SOLUTION BTW
Add and simplify:
3 + 1
10 5
Answer:
I don't under stand the question can u write more clearly
what is the value of 7
7 can be written as :-
7 + 0 = 7
7 - 0 = 7
7 × 1 = 7
7 ÷ 1 = 7
7¹ = 7
_____
Hope it helps ⚜
Which expressions represents "2 less than the product of 5 and a number"?
A) 5n - 2
B) 5/n - 2
C) 2 - 5n
D) 5 + n - 2
Answer:
Option A is correct.Step-by-step explanation:
Let's read the part of the question carefully.(2 less than the product of 5 and a number)
The phrase "2 less" means subtracting 2 from the minuend."the product of 5 and a number" means multiplication of 5 and a number (5x)Conclusion:
Therefore:
Our expression is 5n - 2Hoped this helped.
[tex]GeniusUser[/tex]
A bakery sells two kinds of baked goods: cookies and pie slices. A cookie costs 12 ani a pe sites $4. In one day, the store sold 34 baked goods for a total of $114. How many cookies die itey sele? How many pie slices did they sell? Write a system of equations and solve.
Answer:
uhhhhhhhhhhhh what does this say?
Graph question
f(x)=3/5x-5
two points of function
Step-by-step explanation:
1) plot a point at (0,-5)
2) count 3 units up and 5 units to the right, plot the new point
3) connect the points with a straight line
If there are 10,000 people on a space ship and 7.5 potatoes grow out of 1 potato plant. How many potato plants are we going to need per person?
This can be answered by the equation 10,000 / 7.5 = 1,333.33, so we can round to 1,333.
Hope this helps!
Answer: 1333
Step-by-step explanation:
What is the slope of this line?
a.−23
b.−13
c.13
d.23
Answer:
D
Step-by-step explanation:
I think you meant to put the answers as a fraction
Answer:
if there were slashes in between every answer, the answer would be
D. 2/3
Step-by-step explanation:
m = y2-y1 / x2-x1 (slope formula) ⋅the two points i used were: (0,-3) and (3,-1)
m = -1-(-3) / 3-0 (plug in the points)
m = -1+3 / 3-0
m = 2/3 (Simplify to get your answer)
Justin is buying tickets to a concert online from Ticketmaster. He buys 3 tickets that all have the same price.
There is also a service charge, since he is buying it from Ticketmaster, for $5 per ticket. The total cost of his
order was $138.
Answer:
I don't know if your asking for the price for each ticket, but if you are, it's $41 each.
Step-by-step explanation:
5 x 3 = 15(price of the service charge)
138 - 15 = 123(price with out service tax)
123/3 = 41
Hope this was helpful
Given f (x) = x² + 4x + 5, what is
Answer:
8+h
Step-by-step explanation:
Simply substitute x = 2+h and x = 2 in.
[tex]\displaystyle \large{\frac{f(2+h)-f(2)}{h}=\frac{(2+h)^{2} +4(2+h)+5-[(2)^2+4(2)+5]}{h}}\\\displaystyle \large{\frac{f(2+h)-f(2)}{h}=\frac{4+4h+h^2 +8+4h+5-17}{h}}\\\displaystyle \large{\frac{f(2+h)-f(2)}{h}=\frac{h^2 +8h}{h}=h+8}[/tex]
Therefore, the answer is h+8 or 8+h.
Another way is to differentiate the function with respect to x. This equation or expression is rate of changes from 2 to 2+h with h being any numbers.
The definition of derivative is:
[tex]\displaystyle \large{f\prime(x)= \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}}[/tex]
Notice how both f(2+h)-f(2) over h and the definition of derivative look same except the derivative has limit of h approaching to 0, making h not having any values by default.
Since f(2+h)-f(2) over h does not have limit, we have to add +h when differentiating the function.
[tex]\displaystyle \large{f\prime(x)=2x+4}\\\displaystyle \large{f\prime(2)=2(2)+4}\\\displaystyle \large{f\prime(2)=8}[/tex]
Since f’(2) is 8 but because f(2+h)-f(2)/h does not have limit of h, therefore we add +h.
[tex]\displaystyle \large{\frac{f(2+h)-f(2)}{h}=8+h}}[/tex]
Two angles are complementary. The larger angle is 32° larger than the smaller angle. Find the measure of both angles, and separate your answers with a comma.
The angles are 29° and 61°.
Let the smaller angle be x
Let the larger angle be x+32.
The sum of the complimentary angles will be 90°.
Therefore, x + x + 32° = 90°
2x + 32° = 90°
2x = 90° - 32°
2x = 58°
x = 58/2
x = 29°
The larger angle = x + 32 = 29 + 32 = 61°
Distributive property to remove parentheses 4(3+w)
Answer:
12+4w
Step-by-step explanation:
Multiply 4 by both sides
Solve for g. Make sure to use scrap paper to show your work.
( 144 + 6 x 8 ) ÷ 3 = g
192
64
48
NO NOT PUT THIS IN A FILE
Answer:
64
Step-by-step explanation:
P E D M A S
() ² ÷ × + -
6 x 8 = 48
114 + 48 = 192
(192) ÷ 3 = g
192 / 3 = 64
g = 64
The Hu family goes out for lunch, and the price of the meal is $52. The sales tax on the meal is 6%, and the family also leaves a 20% tip on the pre-tax amount. What is the total cost of the meal?
Answer:
$65.52
Step-by-step explanation:
52 x 6% = 3.12. Multiply the price of the meal by the percentage of sales tax
52 x 20% = 10.4. Multiply the price of the meal by tip percent
$3.12 is the sales tax on the meal
$10.40 is the tip amount
$52 + $3.12 + $10.40 = $65.52
Add sales tax and the tip amount to the price of the meal to find the total
Pre college need withh
Answer:
Step-by-step explanation:
f(9) = |9 + 9| = |18| = 18
f(-4) = |-4 + 9| = |5| = 5
f(0) = |0 + 9| = |9| = 9
pretty boring examples as you're always taking the absolute value of a positive number.
what do you suppose f(-12) is???
f(-12) = |-12 + 9| = |-3| = 3
4x+4 =
What is the answer to this please help
Answer:4(x+1)
Step-by-step explanation:
4x+1
Help plz no links will mark best answer
Answer:
20
Step-by-step explanation:
you know the ratio is 4:3 so you do 4+3=7 then you do 140÷7=20 so I bar represents 20
Okie help help math math math math
Answer: 33
Step-by-step explanation: In the problem, you are given the figure with the following:
2x+3 and 3x+12
Method: Lets set this into an equation. Remember that straight lines add up to 180°.
2x+3+3x+12=180° Combine like terms.
5x+15=180°
5x=165°
x=33°
You can always plug in 33° in the expression that was set to make sure it does give you 180°.
Hope this helps.
Answer:
33
Step-by-step explanation:
Suzanne gets 7% commission on all of her sales. If she makes a sale of $350,000, how much commission will she make?
Answer:
$24,500 of commission
Step-by-step explanation:
350000*0.07 = 24,500
Solve the following system of equations.
x - 2y = -2
7x - 3y = 19
Answer:
Step-by-step explanation:
x-2y=-2 x=2y-2
7x-3y=19
substitute x with 2y-2
7(2y-2)-3y=19 14y-14-3y=19 11y=33 y=3
substitue 3 for y
x-2(3)=-2 x-6=-2 x=4
x=4, y=3
btw if theres more similar questions like this use symbolab a website that solves and show you step by step in solving equations
really useful
The inequality is -2x > 30. What is the solution of the inequality?
Answer:
x < -15
Step-by-step explanation:
Divide both sides by -2.
Next you will reverse the sign because you are dividing by a negative number.
whats 3y/2 if y = -2
Answer:
-3
Step-by-step explanation:
3(-2)/2
-6/2 = -3