Problem 4: Solve the initial value problem
[tex]y' = (y + 1)(y - 2)[/tex]
[tex]y(0) = 1[/tex]
![Problem 4: Solve The Initial Value Problem[tex]y' = (y + 1)(y - 2)[/tex][tex]y(0) = 1[/tex]](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/z8TBmtp4UKmJbaFxHo1ll1Txy26VXxBU.jpeg)
Answers
Separate the variables:
[tex]y' = \dfrac{dy}{dx} = (y+1)(y-2) \implies \dfrac1{(y+1)(y-2)} \, dy = dx[/tex]
Separate the left side into partial fractions. We want coefficients a and b such that
[tex]\dfrac1{(y+1)(y-2)} = \dfrac a{y+1} + \dfrac b{y-2}[/tex]
[tex]\implies \dfrac1{(y+1)(y-2)} = \dfrac{a(y-2)+b(y+1)}{(y+1)(y-2)}[/tex]
[tex]\implies 1 = a(y-2)+b(y+1)[/tex]
[tex]\implies 1 = (a+b)y - 2a+b[/tex]
[tex]\implies \begin{cases}a+b=0\\-2a+b=1\end{cases} \implies a = -\dfrac13 \text{ and } b = \dfrac13[/tex]
So we have
[tex]\dfrac13 \left(\dfrac1{y-2} - \dfrac1{y+1}\right) \, dy = dx[/tex]
Integrating both sides yields
[tex]\displaystyle \int \dfrac13 \left(\dfrac1{y-2} - \dfrac1{y+1}\right) \, dy = \int dx[/tex]
[tex]\dfrac13 \left(\ln|y-2| - \ln|y+1|\right) = x + C[/tex]
[tex]\dfrac13 \ln\left|\dfrac{y-2}{y+1}\right| = x + C[/tex]
[tex]\ln\left|\dfrac{y-2}{y+1}\right| = 3x + C[/tex]
[tex]\dfrac{y-2}{y+1} = e^{3x + C}[/tex]
[tex]\dfrac{y-2}{y+1} = Ce^{3x}[/tex]
With the initial condition y(0) = 1, we find
[tex]\dfrac{1-2}{1+1} = Ce^{0} \implies C = -\dfrac12[/tex]
so that the particular solution is
[tex]\boxed{\dfrac{y-2}{y+1} = -\dfrac12 e^{3x}}[/tex]
It's not too hard to solve explicitly for y; notice that
[tex]\dfrac{y-2}{y+1} = \dfrac{(y+1)-3}{y+1} = 1-\dfrac3{y+1}[/tex]
Then
[tex]1 - \dfrac3{y+1} = -\dfrac12 e^{3x}[/tex]
[tex]\dfrac3{y+1} = 1 + \dfrac12 e^{3x}[/tex]
[tex]\dfrac{y+1}3 = \dfrac1{1+\frac12 e^{3x}} = \dfrac2{2+e^{3x}}[/tex]
[tex]y+1 = \dfrac6{2+e^{3x}}[/tex]
[tex]y = \dfrac6{2+e^{3x}} - 1[/tex]
[tex]\boxed{y = \dfrac{4-e^{3x}}{2+e^{3x}}}[/tex]
Related Questions
One half of a number is subtracted from 25 the answer is 10. What is the number
Answers
Answer:
The answer is 30
Step-by-step explanation:
30÷2=15
25-15=10
Answer:
30
Step-by-step explanation:
25-30/2 = 25-15 = 10 Hope this helps!
Earning Commission
Sales Commision Rate Commission
$450
6%
?
The commission is ?
Answers
Answer:6%
Step-by-step explanation:
This is pretty much asking what the percentage is!
-107
Describe the transformation that maps the pre-image A to the image A.
1)
A)
translated 8 units up and then reflected across the y-axis
B)
- translated 8 units down and then reflected across the y-axis
Eliminate
translated 1 unit to left and then rotated 180° counterclockwise about the
origin
D)
translated 1 unit to right and then rotated 180° counterclockwise about
the origin
Answers
Answer:
C) translated 1 unit to the left and then rotated 180° counterclockwise about the origin
Step-by-step explanation:
by inspection, shape A has been translated 1 unit to the left and then rotated 180° counterclockwise about the origin
The table represents the number of medals won by each member of the Academic Bowl Team. What is the ratio of the number of medals won by Josephine to the total number of medals won by the team?
A) 1/13
B) 1/4
C) 1/3
D) 2/5
Answers
1. Ratio of medals won by Josephine to the total number won by the team: 1/13 (A)
2. Ratio of medals won by Jacob to medals won by Antonio: 2/1 (D)
What is Ratio?Ratio can be described as a quantitative relation that compares how many times a value is contained in another.
1. Ratio of medals won by Josephine to the total number won by the team:
Total won by the team = 26
Medals won by Josephine = 2
Ratio = 2:26 = 2/26
= 1/13 (A)
2. Ratio of medals won by Jacob to medals won by Antonio:
Jacob = 8
Antonio = 4
Ratio = 8:4 = 8/4
= 2/1 (D)
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Can somebody explain how I do this?
Answers
Answer:
You carry the 1 bring the 2 then divide by 3 multiply by 4 times the 2nd power then gimme those points.
Step-by-step explanation:
i have no vlue
Please help me simplify both of these.
Answers
Answer:
A is 3/y, B is 4y
Step-by-step explanation:
qiulian and tammy have 2400 cards. tammy has twice as many cards as qualian.how many cards must tammy gaive qiulian so that tey have a the same number of card?
Answers
Tammy must give Qiulian 800 cards, so that they have the same number of card.
Let Qiulian's card be Q.Let Tammy's card be T.Given the following data:
Total cards, T = 2400 cards.How to solve the word problem.Translating the word problem into an algebraic expression, we have;
Qiulian and Tammy have 2400 cards:
[tex]Q+T=2400[/tex] .....equation 1.
Tammy has twice as many cards as Qiulian:
[tex]T=2Q[/tex] .....equation 2.
To determine the amount of card that Tammy must give Qiulian so that they have the same number of card:
Substituting eqn. 2 into eqn. 1, we have:
[tex]Q+2Q=2400\\\\3Q=2400\\\\Q=\frac{2400}{3}[/tex]
Q = 800 cards.
For Tammy:
[tex]T =2Q\\\\T=2\times 800[/tex]
T = 1600 cards.
Therefore, Tammy must give Qiulian 800 cards, so that they have the same number of card.
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The work shows the first steps of writing a partial fraction decomposition. StartFraction negative 5 x cubed + 2 x squared + 9 x + 2 Over (x squared + 2) squared EndFraction = StartFraction A x + B Over x squared + 2 EndFraction + StartFraction C x + D Over (x squared + 2) squared EndFraction Negative 5 x cubed + 2 x squared + 9 x + 2 = (A x + B) (x squared + 2) + C x + D Negative 5 x cubed + 2 x squared + 9 x + 2 = A x cubed + 2 A x + B x squared + 2 B + C x + D For this partial fraction decomposition, find the unknowns. A = , B = , C = , and D =
Answers
Answer:
I'm certainly late, but it took me a while to figure this one out so I want to leave an answer.
A=-5 B=2 C=19 D=-2
Step-by-step explanation:
I looked back at the video and actually tried to pay attention and the way to figure these problems out is set up all of the degrees of x in an equation. So, take all of the coefficients that are attached to an x^3, and set them up equal to each other. This makes -5=A, which is the first answer. Repeat for x^2 and you have B=2. For x, it becomes 9=2A+C. A is equal to -5 (from the x^3), so you can substitute and get c=19. Repeat for the x^0 term (the constants), and you get 2=2B+D. Plug in the value for B, 2, and get 2=4+D, which then becomes D=-2. That's how you get the answer and I hope this helps somebody.
The values are A=-5, B=2, C=19 and D=-2
What is partial fraction?Partial fractions are the fractions used for the decomposition of a rational expression. When an algebraic expression is split into a sum of two or more rational expressions, then each part is called a partial fraction. Hence, basically, it is the reverse of the addition of rational expressions.
Given:(-5x³+2x²+9x+2) / x²+2 =(Ax + B)/x²+2 + (Cx + d)/x²+2 .......(1)
First, we have to perform the long division as follows
The long division gives
(-5x³+2x²+9x+2 )/ x²+2= ( -5x+2)+ (19x-2)/ x²+2
Now, we have to find the partial fraction for (19x-2)/ x²+2
But, as denominator cannot be factored, So no partial fraction can be calculated.
Thus, (-5x³+2x²+9x+2 )/ x²+2= ( -5x+2)+ (19x-2)/ x²+2
Comparing it with equation (1), we get
Hence, A=-5, B=2, C=19 and D= -2
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How many solutions of linear equation of -x + y = -5
Answers
Answer:
infinite
Step-by-step explanation:
What’s the correct answer for this question? Any help would be appreciated
Answers
Answer:
28.317 litres
62.8 litres
Mathe help #2 lots of points help please
Answers
Medians bisect eachother at centroid in ratio 2:1
ZS=XS=80SU is perpendicular bisector
UY=UZ
So
YZ=2(66)=132Apply Pythagorean theorem
[tex]\\ \tt\hookrightarrow VY^2=XS^2-VS^2=80^2-48^2=6400-2304=4096\implies VY=64[/tex]
Answer:
XS = 80
YZ= 132
VY = 64
Step-by-step explanation:
Firstly, SU is a perpendicular bisector
so, UV= UZ and YZ = 2(66) = 132
Each median bisect each other at centre with the ratio 2 : 1
therefore ZS = 80 = XS
now , by using P.G.T
VY² = XS²-VS²
=> 80²-48²
=> 6400-2304
=> 4096
=> √4096
=> 64
VY = 64
Santino fires a paintball 3 meters from the ground such that the initial trajectory is perfectly horizontal. Assume the paintball does not hit anything in the ground is level. When will the paintball hit the ground?
A. About 3 seconds
B. About 0.78 seconds
C. About 0.52 seconds
D. About 1 second
Answer: B
Answers
a) Multiply 82 X 34 with math workings
b) multiply 69 x 12 with math working
Answers
Multiply each term of 82 by each broken term of 34, i.e.
[tex] \tt \multimap \: 80 \times (30 + 4) + 2 \times (30 + 4)[/tex]
[tex] \tt \nrightarrow \: 80 \times 34 + 2 \times 34[/tex]
[tex] \tt \Rrightarrow \: 2720 + 68[/tex]
[tex] \tt \rightharpoonup \: 2788[/tex]
#b69 = 70-112 = 10+2Multiply each term of 69 by each broken term of 12, i.e.
[tex] \sf \multimap \: 70 \times (10 + 2) - 1(10 + 2)[/tex]
[tex] \sf \nrightarrow \: 70 \times 12 - 1 \times 12[/tex]
[tex] \sf \Rrightarrow \: 840 - 12[/tex]
[tex] \sf \rightharpoondown \: 828[/tex]
Consider the sequence -9, -5, -1, 3, 7,..
1) Is this sequence arithmetic, geometric, or neither?
How can you tell?
2) What is the first term of the sequence?
3) What is the sequence generator?
Answers
Answer:
1) The sequence is arithmetic as the same value is added each time (there is a common difference between terms).
2) First term of the sequence = -9
3) sequence generator = + 4
Step-by-step explanation:
1) The sequence is arithmetic as the same value is added each time (there is a common difference between terms).
2) First term of the sequence = -9
3) sequence generator = + 4
Help Me what is 2,267,742,732,288 in scientific notation for 20 pts
Answers
Answer:
2.267742732288 × 1012
Step-by-step explanation:
Remi plans on spending at most $75 at the mall. She wants to buy a $30 hoodie and some t-shirts that cost $9. Write and solve an inequality to find how many t-shirts Remi can purchase.
Please help! :(
Answers
Explanation:
1. 30+9x less or equal to 75
2. 9x less or equal to 55
3. x less or equal to 55/9
4. she can by 6 t-shirts
Answer:
T-shirts= 5
Step-by-step explanation:
Remi plans on spending 75$ and she already bought a hoodie that cost $30.
x= how many hoodies you buy.
Inequality- $75= 30+75+x
Next- 75-30= $45
Finally- 45/9= 5 t-shirts that you can buy.
Hope that helped! :)
Frank made 105$ for 5 hours of work.
At the same rate, how much would he make for 7 hours of work?
Answers
Answer:
147$
Step-by-step explanation:
105/5 = x/7
x = 105*7/5 = 147
ANSWER FOR 100 POINTS!!! VERY EASY QUESTION, BUT NEED EXPLANATION AS IT's A FREERESPONSE
In the figure below, ∠ABC ≅ ∠DEC and ∠GFE ≅ ∠DCE. Point C is the point of intersection between segment AG and segment BD, while point E is the point of intersection between segment AG and segment DF.
Prove ΔABC ∼ ΔGEF.
Answers
Step-by-step explanation:
We are given that ∠ABC≅∠DEC and ∠GFE≅∠DCE
By Vertical Angles: ∠DEC≅∠GEF and ∠DCE≅∠ACB
By Substitution: ∠ABC≅∠GEF and ∠GFE≅∠ACB
By Angle-Angle Similarity: ΔABC~ΔGEF
Hope this helps! Sorry if I didn't explain well :(
Question 9 of 20
Mr. Chen is parking in a lot downtown. The graph shows the relationship
between the time and the total cost of parking . Evaluate the function for an
input of 10
Cost of Parking in Lot
20
l
A. After 18 hours, the cost is $18.
B. After 18 hours, the cost is $10.
C. After 10 hours, the cost is $18.
D. After 10 hours, the cost is $10.
Answers
The answer to the problem(i did this question before) B
The solution to the question by reading the graph is:
When x(Time of parking) is 6 hours, y(Cost of Parking) is 12 hours.
What is graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
here, we have,
Mr. Chen is parking in a lot downtown. The graph shows the relationship
between the time and the total cost of parking .
now, we have to find the function for an input of 10 Cost of Parking in Lot 20.
Hence, The solution to the question by reading the graph is:
When x(Time of parking) is 6 hours, y(Cost of Parking) is 12 hours.
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Please help! I will give Brainliest! A Bernoulli counting process has the following realization:
N0=0, N1=1, N2=1, N3=1, N4=2, N5=3, N6=3, N7=4, N8=4, N9=5
What is the value of X6?
What is the value of S3?
If the success probability is p=0.2, What is P{N2=1, N5=3, N7=4}?
Answers
The Bernoulli counting process depicts that the value of X6 will be 0.
How to solve the Bernoulli counting processThe value of X6 will be:
= N6 - N5
= 3 - 3
= 0
Also, the value of S3 will be:
X1 = 1 - 0 = 1
X2 = 1 - 1 = 0
X3 = 1 - 1 = 0
Therefore, S3 will be:
= X1 + X2 + X3
= 1 + 0 + 0
= 1
In conclusion, the value of S3 is 1.
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need help with This question
Answers
Answer:
56 meters
none of these
Step-by-step explanation:
4*2=8
8*6=48
48+8=56
Answer:
32
Step-by-step explanation:
you add all the sides and to find the missing side you add 4+4 because it is the opposite and the other 8-6 to get 2 so 4+8+8+6+4+2=32
Car A is traveling west at 40 mph and car B is traveling north at 55 mph. Both are departing from the intersection of two roads at the same time. Write a function representing the distance between the two cars in terms of the time , in hours, that has elapsed since the cars left the intersection.
Answers
Answer:
[tex]5\sqrt{185}t[/tex]
Step-by-step explanation:
after time t (hr), the distance between A and starting point is 40t
the distance between B and starting point is 55t
These two distances are perpendicular to each other, so
the distance between A and B is
[tex]\sqrt{(40t)^2+(55t)^2}=\sqrt{4625t^2}=5\sqrt{185}t[/tex]
A computer generates 50 integers from 1 to 8 at random. The results are recorded in this table.
Outcome 1 2 3 4 5 6 7 8
Number of times outcome occurred
5 8 9 7 4 6 5 6
What is the experimental probability of the computer generating a 4 or a 7?
8%
14%
22%
24%
Answers
Answer:
D. 24%Step-by-step explanation:
Number of 4 generated is 7 and number of 7 generated is 5.
Total of 7 + 5 = 12 outcomes out of 50:
P(4 or 7) = 12/50 = 24/100 = 24%The experimental probability of the computer generating a 4 or a 7 is D. 24%.
What is experimental probability?Experimental probability calculates the probability of some event from the results of experiments.
For an event E, we get the experimental probability of that event
[tex]P_e(E) = \dfrac{\text{Number of times E occurred}}{\text{Number of times experiments was done}}[/tex]
where P_e(E) is denoting the experimental probability of occurrence of E.
A computer generates 50 integers from 1 to 8 at random. The results are recorded in this table.
Outcome 1 2 3 4 5 6 7 8
Number of times outcome occurred
5 8 9 7 4 6 5 6
The Total of 7 + 5 = 12 outcomes out of 50:
The experimental probability of the computer generating a 4 or a 7
P(4 or 7) = 12/50
= 24/100
= 24%
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bernice buys 5 VHS tapes. each VHS tape cost $9. if bernice receives a $2 discount on each vhs, what is the total amount of money she spends? write an expression that matches the words.
Answers
Answer: bernice buys 5 VHS tapes. each VHS tape cost $9. if bernice receives a $2 discount on each vhs, what is the total amount of money she spends? write an expression that matches the words.
the answer is $11
Step-by-step explanation: 9+2=11
Find the measure of angle
Answers
Answer:
It can be any measure
Step-by-step explanati
The solutions to a certain quadratic equation are x= -4 and X = 3. Write the equation in standard form below.
[blank] +x [blank] [blank] = 0
Answers
Answer:
x² + x - 12 = 0
Step-by-step explanation:
If x = -4 and x = 3 are solutions to a quadratic equation, then
(x + 4)(x - 3) = 0
Expand the brackets: x² - 3x + 4x - 12 = 0
Combine like terms: x² + x - 12 = 0
Find the value of x.
Answers
Answer:
x=56
Step-by-step explanation:
Hope this helps!
Answer:
x = 12.05
using sine rule:
[tex]\boxed{sin(a) = \frac{opposite}{hypotenuse} }[/tex]
Here given the angle is 35° and hypotenuse is 21.
To find x:
→ sin(35) = [tex]\frac{x}{21}[/tex]
→ x = 21 * sin(35)
→ x = 12.04510516
→ x = 12.05 ................rounded to nearest tenth.
THIS IS DUE IN 5 MINUTES!!!!!!!!!!! I NEEED THIS ASAP! A 13 3/4 -inch candle burns down in 11 hours. If b represents how much of the candle, in inches, has burned away at any time given in hours, t, write a proportional equation for b in terms of t that matches the context.
Answers
I HOPE THIS HELPSSSSSS
GOOD LUCKKKKKK :)
Find the values of x, y, and z in the
parallelogram. Note that the figure is not
drawn to scale.
Answers
Step-by-step explanation:
we know that sum of all angles of a triangle is 180°
so, angle N + angle O + angle M = 180°
=> (-5b+16)+(-b+8)+(-9b-9) = 180°
=> -5b+16-b+8-9b-9 = 180°
=> -15b+15 = 180°
=> -15b = 180-15
=> -15b = 165
=> b = -165/15
=> b = -11
angle N => -5(-11)+16
=> 55+16 => 71
angle O => -(-11)+8
=> 11+8 => 19
angle M => -9(-11)-9
=> 99-9
=> 90
8x18divided by (16divide8x3)
Answers
Answer:
It might be 96. I don't know
Step-by-step explanation: