Find a general solution for y′′−4y′+4y=0;y(0)=2,y′(0)=4.
Answers
The general solution for the differential equation y′′−4y′+4y=0, with initial conditions y(0)=2 and y′(0)=4, is y(x) = (2 + 2x)e^(2x).
To find the general solution of the given differential equation, we can assume that y(x) can be expressed as a power series, y(x) = Σ(a_nx^n), where a_n are constants to be determined. Differentiating y(x), we get y′(x) = Σ(na_nx^(n-1)) and y′′(x) = Σ(n(n-1)a_nx^(n-2)). Substituting these expressions into the differential equation, we obtain the power series Σ(n(n-1)a_nx^(n-2)) - 4Σ(na_nx^(n-1)) + 4Σ(a_nx^n) = 0. Simplifying the equation and setting the coefficients of each power of x to zero, we find that a_n = (n+2)a_(n+2)/(n(n-1)-4n) for n ≥ 2. Using this recursive relationship, we can determine the values of a_n for any desired term in the power series.
Given the initial conditions y(0)=2 and y′(0)=4, we can substitute these values into the power series representation of y(x) and solve for the constants. By doing so, we find that a_0 = 2, a_1 = 6, and all other coefficients are zero. Thus, the general solution is y(x) = (2 + 2x)e^(2x), which satisfies the given differential equation and initial conditions.
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Related Questions
find y intercept
y=50+5x
answer needs to be a point ex. (0,0)
Answers
Answer:
(0, 50)
Step-by-step explanation:
The equation of a line is in the form of the y-intercept form
\(y = mx + b\)
where m is the slope and B is the y intercept. So in our case
\(y = 50 + 5x\)
if we use the commutative property we can rewrite our equation as
\(y = 5x + 50\)
therefore b = 50 is our why intercept
In slope-intercept form, y = mx + b, where m = slope, and y-intercept = b:
b represents the y-coordinate of the y-intercept (0, b). Hence, in your given equation: y =5x + 50:
b = 50, and its corresponding x-coordinate is 0.
The y-intercept is (0, 50).
find the solution of given expression
\( \sqrt{587 \times 587} \)
Answers
Answer:
√{587×587}=√587²=±587 is your answer
Answer:
587
Step-by-step explanation:
If you look at the problem carefully, you will see that the question is basically just asking "what is the square root of 587 squared?" The square root and squared of something always cancel out, so the answer would be 587.
You could also just use your calculator. If you don't have one, your phone or computer should. Just plug it in and it will solve for you.
Hope this information helps. Please let me know if you need any more help!
determine the derivatives of the following inverse trigonometric functions:
(a) f(x)= tan¹ √x
(b) y(x)=In(x² cot¹ x /√x-1)
(c) g(x)=sin^-1(3x)+cos ^-1 (x/2)
(d) h(x)=tan(x-√x^2+1)
Answers
To determine the derivatives of the given inverse trigonometric functions, we can use the chain rule and the derivative formulas for inverse trigonometric functions. Let's find the derivatives for each function:
(a) f(x) = tan^(-1)(√x)
To find the derivative, we use the chain rule:
f'(x) = [1 / (1 + (√x)^2)] * (1 / (2√x))
= 1 / (2x + 1)
Therefore, the derivative of f(x) is f'(x) = 1 / (2x + 1).
(b) y(x) = ln(x^2 cot^(-1)(x) / √(x-1))
To find the derivative, we again use the chain rule:
y'(x) = [1 / (x^2 cot^(-1)(x) / √(x-1))] * [2x cot^(-1)(x) - (x^2 + 1) / (x(x-1)) - 1 / (2√(x-1))]
Simplifying further:
y'(x) = 2 cot^(-1)(x) - (x^2 + 1) / (x(x-1)) - 1 / (2√(x-1))
Therefore, the derivative of y(x) is y'(x) = 2 cot^(-1)(x) - (x^2 + 1) / (x(x-1)) - 1 / (2√(x-1)).
(c) g(x) = sin^(-1)(3x) + cos^(-1)(x/2)
To find the derivative, we apply the derivative formulas for inverse trigonometric functions:
g'(x) = [1 / √(1 - (3x)^2)] * 3 + [-1 / √(1 - (x/2)^2)] * (1/2)
Simplifying further:
g'(x) = 3 / √(1 - 9x^2) - 1 / (2√(1 - x^2/4))
Therefore, the derivative of g(x) is g'(x) = 3 / √(1 - 9x^2) - 1 / (2√(1 - x^2/4)).
(d) h(x) = tan(x - √(x^2 + 1))
To find the derivative, we again use the chain rule:
h'(x) = sec^2(x - √(x^2 + 1)) * (1 - (1/2)(2x) / √(x^2 + 1))
= sec^2(x - √(x^2 + 1)) * (1 - x / √(x^2 + 1))
Therefore, the derivative of h(x) is h'(x) = sec^2(x - √(x^2 + 1)) * (1 - x / √(x^2 + 1)).
These are the derivatives of the given inverse trigonometric functions.
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3/4 + 1/2 + 1/3 + 2/9 + ....... = 9/4 es verdadero oh falso y como se resuelve ?
Answers
The missing fraction is 7/9
Finding missing values in an addition equationThe given equation is:
\(\frac{3}{4}+\frac{1}{2}+\frac{2}{9}+x=\frac{9}{4}\)
This can be simplified as:
\(\frac{27+18+8+36x}{36} =\frac{9}{4} \\\\\frac{53+36x}{36} =\frac{9}{4}\)
Cross multiply
\(9(36)=4(53+36x)\\\\324=212+144x\\\\144x=324-212\\\\144x=112\\\\x=\frac{112}{144} \\\\x=\frac{7}{9}\)
Therefore, the missing value is \(\frac{7}{9}\)
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two fair dice, each with at least 6 faces are rolled. on each face of each dice is printed a distinct integer from 1 to the number of faces on that die, inclusive. the probability of rolling a sum of 7 is 3 4 of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is 1 12 . what is the least possible number of faces on the two dice combined?
Answers
Using the Counting the faces of two dice,
the atleast 17 number of faces are possible on the combination of two dice.
First, we have to count the favorable cases,
We know both dice have atleast 6 faces.
It gives 6 favorable cases for a sum of 7.
If we can count them if mean even if dice had more than 6 faces, will matter of for sum of 7.
Now, the mean, we need 6× 4/3=8 (favorable cases for the sum of 10)
If we count 3 favorable cases for each having 6 faces.
For more than 9 faces on a dice matter give the denominator (sample space) are the same for both sum of 7 and the sum of 10, and the probability of one is in proportion to the other.
It mean, additional 5 cases must be ( 7,2), (2,7), (8,2), (2,8) ,(9,1)
So , one of the dice has 8 faces and the other has atleast 9 faces.
Now, we must have atleast 17 combined faces of two dice for probability . So, we get if 12 with configration of 8 faces on one dice.
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Plz help due tomorrow
Answers
Answer: $21000
Step-by-step explanation:
By uwing option A, the salesperson will make (40 × $21) = $840.
Let the amount of sales that is needed for the week to be thesame as option A be x. Therefore,
4% of x = $840
4% × x = $840
0.04x = $840
x = $840/0.04
x= $21000
The salesperson need to sell $21,000 worth of sales
two numbers have ratio 12:5. Their difference is 98. Find the larger
number.
Answers
Identify whether the relationship between a and b in the image below is complementary, or linear pair/supplementary.
I’m thinking it’s complementary?
Answers
Answer:
Complementary angles.
Step-by-step explanation:
Complimentary angles means sum of angles = 90°
Supplementary angles of linear pair means sum of pair of angles = 180°
As shown in the picture attached, sum of both the angles 'a' and 'b' is equal to 90°,
a + b = 90°
Angles 'a' and 'b' will be complementary angles.
If f(x)= x^2 lnx, then f ‘(x) = ___
Answers
The derivative of f(x) = x^2 ln(x) is given by f'(x) = 2x ln(x) + x.
To find the derivative of f(x), we can use the product rule, which states that if we have a function f(x) = g(x) * h(x), then the derivative of f(x) with respect to x is given by f'(x) = g'(x) * h(x) + g(x) * h'(x).
In this case, g(x) = x^2 and h(x) = ln(x). Applying the product rule, we have:
f'(x) = (2x * ln(x)) + (x * (1/x))
= 2x ln(x) + 1.
Therefore, the derivative of f(x) = x^2 ln(x) is f'(x) = 2x ln(x) + x.
To find the derivative of f(x) = x^2 ln(x), we need to apply the product rule. The product rule is a rule in calculus used to differentiate the product of two functions.
Let's break down the function f(x) = x^2 ln(x) into two separate functions: g(x) = x^2 and h(x) = ln(x).
Now, we can differentiate each function separately. The derivative of g(x) = x^2 with respect to x is 2x, using the power rule of differentiation. The derivative of h(x) = ln(x) with respect to x is 1/x, using the derivative of the natural logarithm.
Applying the product rule, we have f'(x) = g'(x) * h(x) + g(x) * h'(x).
Substituting the derivatives we found, we get f'(x) = (2x * ln(x)) + (x * (1/x)). Simplifying the expression, we have f'(x) = 2x ln(x) + 1.
Therefore, the derivative of f(x) = x^2 ln(x) is f'(x) = 2x ln(x) + x.
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Who is correct pls answer
Answers
I’d say neither are correct because a prism looks like this...
Katie would be correct if she had lines within the L shaped area
the are of one of the bases of the figure shown above is about 21.2 cm sq. What is the surface area, in cm sq, of the figure? round your answer to the nearest tenth 9,7,7,7 cm
Answers
Answer:
105.4
Step-by-step explanation:
I got the answer wrong when I answered 95 for some reason and it told me. the answer was 105.4
The surface area is 105.4 square centimeters.
What is surface area?A surface area formula is a mathematical approach to discovering the total place of any 3-dimensional object occupied by means of all of its surfaces.
what are surface area examples?The surface area is the overall location blanketed by way of all of the faces of a 3-d item. For an instance, if we need to discover the quantity of paint required to paint a dice, then the surface on which the paint can be carried out is its floor place. it is continually measured in rectangular units.
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The volume of a solid revolution generated by rotating the curve y = f(x) and x = f(y) between x = a and x = b , y = a and y = b through 360 degrees about the x-axis and y-axis is given
\(V_{x} =\int\limits^b_a {\pi y^{2} } \, dx\) and \(V_{y} =\int\limits^b_a {\pi x^{2} } \, dy\)
The diagram shows the line y = 1, the line y = 4 and part of the curve \(y=3x^2\). The shaded region is rotated through 360 degrees about the y-axis. Find the exact value of the volume of revolution obtained. Leave your answer in pi.
Answers
Answer:
\(\dfrac{5}{2}\pi\)
Step-by-step explanation:
Rotation about the y-axis
\(\textsf{Volume}=\displaystyle \int^b_a \pi x^2\:\text{d}y\)
where:
b = upper limita = lower limitx is a function of yGiven function of y: \(y = 3x^2\)
Rewrite the given function as a function of y:
\(\implies x^2=\dfrac{1}{3}y\)
Substitute the values into the formula:
\(\implies \displaystyle \int^4_1 \dfrac{1}{3}\pi y\:\:\text{d}y\)
\(\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int ay^n\:\text{d}y=a \int y^n \:\text{d}y$\end{minipage}}\)
\(\boxed{\begin{minipage}{4 cm}\underline{Integrating $y^n$}\\\\$\displaystyle \int y^n\:\text{d}y=\dfrac{y^{n+1}}{n+1}+\text{C}$\end{minipage}}\)
Take out the constant and integrate:
\(\begin{aligned}\implies \dfrac{1}{3}\pi\displaystyle \int^4_1 y\:\:\text{d}y & = \dfrac{1}{3}\pi \left[\dfrac{1}{2}y^2\right]^4_1\\\\& =\dfrac{1}{3}\pi \left[\dfrac{1}{2}(4)^2-\dfrac{1}{2}(1)^2\right]\\\\&=\dfrac{1}{3}\pi\left[8-\dfrac{1}{2}\right]\\\\&=\dfrac{5}{2}\pi \end{aligned}\)
Therefore, the exact value of the volume of revolution is:
\(\dfrac{5}{2}\pi\)
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help me with this please
Answers
suppose the data on natural birth weights (collected by the world health organization (who)) is normally distributed with the mean equal to 7.25 pound for a full-term birth and standard deviation of 1.0 pounds. what is the probability that a randomly selected baby will weigh more than 9.75 pounds at birth? 0.0062
Answers
The probability that a randomly selected baby will weigh more than 9.75 pounds at birth is 0.0062, or about 0.62%. This means that the vast majority of babies will weigh less than 9.75 pounds at birth, as this value is more than two standard deviations above the mean.
We can use the standard normal distribution to solve this problem by first standardizing the value of 9.75 pounds using the formula:
z = (x - mu) / sigma
where x is the value of 9.75 pounds, mu is the mean of 7.25 pounds, and sigma is the standard deviation of 1.0 pound.
Substituting the values, we get:
z = (9.75 - 7.25) / 1.0 = 2.5
We can then use a standard normal distribution table or calculator to find the probability of a z-score greater than 2.5. From the table or calculator, we find that this probability is approximately 0.0062.
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francine wants to join a health club and has narrowed it down to two choices. the sportshaus charges an initiation fee of $ 500 and $ 10 per month. fitness first has an initiation fee of $ 50 and charges $ 25 month.
Answers
The equation for the total cost for Sports Haus is 500 + 10x and the equation for Fitness First is 50 + 25x .
In the question ,
it is given that ,
Francie wants to join the health Club .
For the Sports Haus the initiation fees = $500
per month charge = $10
let the number of months be= "x" ,
so , the equation to represent the total cost for Sports Haus Club for x months is 500 + 10x .
For the Fitness First the initiation fees = $50
per month charge = $25
so , the equation to represent the total cost for Fitness First Club for x months is 50 + 25x .
Therefore , The equation for the total cost for Sports Haus is 500 + 10x and the equation for Fitness First is 50 + 25x .
The given question is incomplete , the complete question is
Francine wants to join a health club and has narrowed it down to two choices. The Sports Haus charges an initiation fee of $ 500 and $ 10 per month. Fitness First has an initiation fee of $ 50 and charges $ 25 month . write an equation for the total cost of each health club for x months .
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can someone help me find the equation!
Answers
Which is the equation of the given line? (2,4) (2,-7)
A) x=2
B) y=2
C) x=4
D) y=2x
Answers
-11
Answer: D) y=2x-11
Consider the forced spring-mass system: d²y/dt² + 5y = 8sin(at) where a is a parameter a. For which value of a does the system exhibit resonance? b. Find the general solution for the a found in (a)
Answers
The forced spring-mass system is described by the differential equation d²y/dt² + 5y = 8sin(at), where a is a parameter.
To determine the value of a for which the system exhibits resonance, we need to find the frequency at which the driving force matches the natural frequency of the system. The natural frequency is given by ω₀ = √(k/m), where k is the spring constant and m is the mass.
In this case, the natural frequency is √(5), since the coefficient of y in the differential equation is 5. Resonance occurs when the frequency of the driving force matches the natural frequency, so we set ω₀ = a and solve for a. Hence, a = √(5).
For the value of a found in part (a), which is a = √(5), the general solution of the forced spring-mass system can be obtained by using the method of undetermined coefficients. The complementary solution is y_c(t) = c₁cos(√(5)t) + c₂sin(√(5)t), where c₁ and c₂ are arbitrary constants.
To find the particular solution, we assume y_p(t) = Asin(√(5)t) + Bcos(√(5)t), where A and B are coefficients to be determined. Substituting this into the differential equation and solving for A and B, we can find the general solution for the given value of a.
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Evaluate.
53−(64÷8)+102
Enter your answer in the box.
I think the answer 217. But I don't know if am correct or not.
Thank you!
Answers
Hey! First we have to solve what's in parentheses. We can use the PEMDAS method too. P = parentheses E = exponents M = multiplication D = division A= addition S = subtraction. We also solve left to right. So starting off with what's in the parentheses, 64/8, that equals 8. So now the equation is 53-8+102.
Now is 53 - 8, which is 45. Now we just add 102 and 45. Your final answer would be 147! Hope this helped.
If S(0,6) and T(-10,-1) are the endpoints of directed line segment ST. What is the y coordinate of the point that divides the segment into a ratio of 3:4
Answers
Answer:
The y coordinate is 3
Step-by-step explanation:
Here, we want to find the y-coordinate of the line that divides the line segment in the given ratio.
To calculate this, we shall be using the internal division formula;
It is as follows;
( (mx2 + nx1) /m + n , (my2 + ny1)/( m + n))
In this question;
m = 3 and n = 4
x1 = 0 , y1 = 6
x2 = -10, y2 = -1
Now substituting these values into the equation, we have;
(3(-10) + 4(0)/(3 + 4) , (3(-1) + 4(6)/(3+4))
= -30/7 , (-3 + 24)/7
= -30/7 , 21/7
= -30/7 , 3
Given: AB=AC, AD=AE and DC=EB
Prove: <1 is equal to <2
PLEASE HELP! Thank you!
Answers
The prove of ∠1 equal to ∠2 is shown below.
What is mean by Triangle?
A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.
Given that;
⇒ AB = AC, AD = AE and DC = EB
Now,
In ΔDBC and ΔEBC;
⇒ DC = EB (Given) ..(i)
Since, AB = AC
So, the ΔABC is a isosceles triangle.
Hence, ∠DBC = ∠ ECB ..(ii)
Here, AB = AC and AD = AE
So, BD = EC ..(iii)
Hence, By the condition (i), (ii) and (iii);
By SAS condition,
⇒ ΔDBC ≅ ΔEBC
Hence, We get;
⇒ ∠1 = ∠2
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50 POINTS. Select the correct answer from each drop-down menu.
15
In AABC, m B = 90°, COS(I) = 17, and AB = 16 units.
V
V
units.
Based on this information, mzA= v ,mzc=
o, and AC =
Note that the angle measures are rounded to the nearest degree.
Answers
Answer:
m∠C=28°, m∠A=62°, AC=34.1 units
Step-by-step explanation:
Given In ΔABC, m∠B = 90°, , and AB = 16 units. we have to find m∠A, m∠C, and AC.
As,
⇒
By angle sum property of triangle,
m∠A+m∠B+m∠C=180°
⇒ m∠A+90°+28°=180°
⇒ m∠A=62°
Now, we have to find the length of AC
⇒
The length of AC is 34.1 units
A tire with a radius of 22 centimeters
Answers
138.28
22/7 x 22 = 968/7 = 138.28
Mr. Tanaka is loading his boat with many identical 45-pound boxes. Mr. Tanaka weighs 190 pounds. The boat can carry at most 1,000 pounds. If each box is fully-filled, how many boxes can he load on the boat?
Write an inequality that represents this situation.
Answers
Answer:
there should be 14, 45 pound boxes
Step-by-step explanation:
1000-190= 810
810/45=14 boxes
A runing track is 400m long. Carrie runs wight laps. How far is this in kilometres.
Answers
Multiply length of track by number of times run:
8 x 400 = 3,200 meters
1 km = 1000 meters
3200 meters / 1000 = 3.2 kilometers
Answer: 3.2 kilometers
the value of sin^2 30° - cos^2 30° is
Answers
Answer:
-1/2
Step-by-step explanation:
Im not going to go into detail about how to compute sin30 and cos 30, but know that sine and cosine are just the ratios between the sides of a 30-60-90 triangle with a hypothenuse of 1.
With that being said:
sin^2 30 - cos^2 30
= (1/2)^2 - ((sqrt3)/2)^2
= 1/4 - 3/4
= -1/2
Answer:
-1/2
Step-by-step explanation:
What is an interval of increase ?
Answers
The first steps in determining the perimeter of triangle ABC are shown.
To the nearest whole unit, what is the perimeter of triangle ABC?
6 units
8 units
14 units
16 units
Answers
Answer:
C) 14 Units
Step-by-step explanation:
E2020
Please ASAP Help
Will mark brainlest due at 12:00
Answers
Answer:
1
Step-by-step explanation:
we just find the middle of the 2 places
The urface area of a cylinder i 24π m². The ditance around the bae of the cylinder i 3π meter and the diameter of a bae of the cylinder i 4 meter. What i the height of the cylinder? Enter your anwer, a a implified fraction, in the box
Answers
After solving, the height of the cylinder is 6 meters.
The surface area of a cylinder is 24π m².
The distance around the base of the cylinder is 3π meter.
The diameter of a base of the cylinder is 4 meter.
Now the radius of the cylinder = diameter/2
The radius of the cylinder = 4/2
The radius of the cylinder = 2 meter.
The formula of surface area of cylinder:
S = 2πrh
From the question:
2πrh = 24π
Now putting the value of r
2π × 2 × h = 24π
4πh = 24π
Divide by 4π on both side, we get
h = 6
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The complete question is:
The surface area of a cylinder is 24π m². The distance around the base of the cylinder is 3π meter and the diameter of a base of the cylinder is 4 meter. What is the height of the cylinder? Enter your answer, in a simplified fraction, in the box.