Solve the given initial-value problem. y'' + 4y' + 4y = (5 + x)e−2x, y(0) = 3, y'(0) = 9
y(x) =
Answers
To solve the given initial-value problem, first find the complementary solution, then the particular solution, and finally apply the initial conditions.
The complementary solution is obtained by solving the homogeneous equation y'' + 4y' + 4y = 0. The characteristic equation is r^2 + 4r + 4 = 0, which can be factored as (r + 2)^2 = 0. The root is r = -2, which is a repeated root, so the complementary solution is y_c(x) = C_1e^(-2x) + C_2xe^(-2x).
Next, find the particular solution y_p(x) for the non-homogeneous equation y'' + 4y' + 4y = (5 + x)e^(-2x). Using the method of undetermined coefficients, let y_p(x) = A(x)e^(-2x) + B(x)e^(-2x). Differentiate y_p(x) and substitute it into the non-homogeneous equation. Then, solve for A(x) and B(x) to obtain y_p(x).
Finally, combine y_c(x) and y_p(x) to form the general solution y(x) = y_c(x) + y_p(x). Apply the initial conditions y(0) = 3 and y'(0) = 9 to find the constants C_1 and C_2. Substitute the values of C_1 and C_2 back into the general solution to get the final answer.
The initial-value problem is given byy'' + 4y' + 4y = (5 + x)e−2x, y(0) = 3, y'(0) = 9.To solve the given initial-value problem, we need to determine the roots of the characteristic equation of the equation y'' + 4y' + 4y = 0.The characteristic equation of the equation is given byr² + 4r + 4 = 0Solving for r,r = -2 (twice)Since the roots of the characteristic equation are equal, the solution to the homogeneous equation is given byy = (c₁ + c₂x)e^(-2x)We need to find the particular solution to the non-homogeneous equation.The right-hand side of the non-homogeneous equation is of the form (5 + x)e^(-2x). This suggests that the particular solution has the formy_p = (Ax + B)e^(-2x)Differentiating the above equation twice, we gety'_p = (-2Ax - 2B + A)e^(-2x)y''_p = (4Ax + 4B - 3A)e^(-2x)Substituting the above equations into the non-homogeneous equation, we get(4Ax + 4B - 3A)e^(-2x) + 4(-2Ax - 2B + A)e^(-2x) + 4(Ax + B)e^(-2x) = (5 + x)e^(-2x)Simplifying, we getA = -1, B = 2Thus, the particular solution is given byy_p = -(x + 2)e^(-2x)The general solution to the non-homogeneous equation is given byy = (c₁ + c₂x)e^(-2x) - (x + 2)e^(-2x)The solution to the given initial-value problem is obtained by substituting the initial values y(0) = 3 and y'(0) = 9 into the above equation.y(0) = (c₁ + c₂ × 0)e^(-2 × 0) - (0 + 2)e^(-2 × 0) = 3Hence,c₁ - 2 = 3c₁ = 5Similarly, y'(0) = -2c₁ + 4c₂ + 2 = 9c₁ - 2c₂ = 7Solving for c₁ and c₂, we getc₁ = 5, c₂ = -1Thus, the solution to the given initial-value problem is given byy = (5 - x)e^(-2x) - (x + 2)e^(-2x)y(x) = (5 - x - x - 2)e^(-2x)y(x) = (3 - 2x)e^(-2x)Hence, y(x) = (3 - 2x)e^(-2x) is the required solution.
Learn more about equation, here: brainly.com/question/29657983
#SPJ11
Related Questions
use logarithmic differentiation to find the derivative of y = 3 √x(x −2) / x^2 1 . leave your answer unsimplified.
Answers
To use logarithmic differentiation to find the derivative of y = 3√x(x −2) / x^2, we first take the natural logarithm of both sides:
ln(y) = ln(3√x(x −2) / x^2)
Then we use the logarithmic differentiation rule, which states that if y = f(x) is a function of x, then
y' / y = (ln(f(x)))'
Using this rule, we can find the derivative of ln(y) and simplify it:
ln(y) = ln(3) + (1/2)ln(x(x-2)) - 2ln(x)
ln(y)' = 0 + (1/2)(1/(x(x-2)))(2x-2) - 2*(1/x)
ln(y)' = (x-1)/(x(x-2))
Now we can find y' by multiplying both sides of the original equation by y and substituting in the expression we just found for ln(y)':
y = 3√x(x −2) / x^2
ln(y) = ln(3) + (1/2)ln(x(x-2)) - 2ln(x)
y' / y = (x-1)/(x(x-2))
y' = y*(x-1)/(x(x-2))
Substituting the original expression for y, we have:
y' = (3√x(x −2) / x^2)*((x-1)/(x(x-2)))
Therefore, the derivative of y = 3√x(x −2) / x^2 is y' = (3√x(x −2) / x^2)*((x-1)/(x(x-2))).
To know more about logarithmic differentiation, visit the link given below:
https://brainly.com/question/30766127
#SPJ4
If Rich decides to make no payments during the 4. 5 years, the interest will be capitalized at the end of that period. A. What will the new principal be when he begins making loan payments? b. How much interest will he pay over the llfe of the loani 1. Suppose Rich only paid the interest during his 4 years in school and th
Answers
a) The new principal (principal + capitalized interest) when Rich begins making loan payments will be $9,425.10.
b) The total interest that Rich will pay over the life of the loan is $4,914.20.
Simple interest (S.I.) is a method of calculating the amount of interest on a specific principal amount at a certain rate. For example, when one takes out a loan of Rs. 5000, at the rate of 10 p.A. Two years The person's two years of interest will be the money borrowed by SI.
Interest on student loans is based on simple interest. If the student does not pay interest, the only interest capitalized is that for the period during which the payment was not made.
Some students choose to pay interest during the non-repayment period to reduce the capitalized interest and principal during the repayment period.
Capitalization refers to the addition of interest to the principal amount.
Period of student loan = 10 years
The amount of the federal unsubsidized student loan = $7,900
Interest rate = 4.29%
Capitalized interest during the non-payment time = $1,525.10 ($7,900 x 4.29% x 4.5)
New principal when Rich begins making loan payments = $9,425.10 ($7,900 + $1,525.10)
Total interest for loan = $4,914.20 ($7,900 x 4.29% x 14.5)
Complete Question:
Rich is attending a 4-year college. As a freshman, he was approved for a 10-year, federal unsubsidized student loan in the amount of $7,900 at 4.29%. He knows he
has the option of beginning repayment of the loan in 4.5 years. He also knows that during this non-payment time, interest will accrue at 4.29%.
If Rich decides to make no payments during the 4.5 years, the interest will be capitalized at the end of that period.
a. What will the new principal be when he begins making loan payments?
b. How much interest will he pay over the life of the loan?
Learn more about Payment:
https://brainly.com/question/30296905
#SPJ4
The safety rules for a playground state that the height of the slide and the distance from the base of the ladder to the front of the slide must be in a ratio of 3 : 5. What would be the length of this slide? Round your answer to the nearest whole number. ____feet
Answers
Answer: 12
Step-by-step explanation: c = √a2 + b2
= √(6)2 + (10)2
= √36 + 100
= √136
= 2√34
= 11.661903789691
Round up
12
The length of the slide has a height of 6 feet and the base of 10 feet would be equal to the nearest whole number 12 feet.
The ratio of the height of the slide and the distance from the base of the ladder to the front of the slide is 3 : 5.
The height of the slide is 6 feet.
The base of the ladder is 10 feet.
What is Pythagoras Theorem?The Pythagoras is the sum of the square of two sides which is equal to the square of the longest side.
From the Pythagoras Theorem,
\(\rm a^{2} =b^{2} +c^{2} \\\rm a^{2} =10^{2} +6^{2} \\\rm a^{2} =100+36\\\rm a = \sqrt{136} \\\rm a = 11.6619\\\rm a=12\)
By round off up to nearest whole number. The length of the slide would be 12 feet.
Learn more about the Pythagoras Theorem here:
https://brainly.com/question/343682
Find the length of the third side to the nearest tenth.
3
9
Answers
Answer:
8.5
Step-by-step explanation:
X^2 = 9^2 - 3^2
x^2 = 81 - 9
x^2 = 72
x = √72
x = 8.485
A carpenter is making a wooden window frame that has a width of 1 inch.
A window with a one-inch frame is shown. The frame is comprised of a rectangle and a semicircle. The rectangle has side lengths of 12 inches and 48 inches. The semicircle has a radius of 6 inches. The frame is 1-inch wider than the window.
How much wood does the carpenter need to build the frame?
5.5 + 106 square inches
5.5 + 116 square inches
5.5 + 153 square inches
5.5 + 162 square inches
Answers
Answer:
5.5 + 106 square inches
Step-by-step explanation:
It's the first option.
Answer:
A. 5.5 + 106 square inches
Step-by-step explanation:
L is the circle with the equation x²+y²=9
full question in photo :)
Answers
The values of the variables, a, b, and c obtained from the equation of the circle and the coordinates of the point P are;
a) a = 2
b = -2
c = 4
What is the general equation of a circle?The general equation of a circle is; (x - h)² + (y - k)² = r²
Where;
(h, k) = The coordinates of the center of the circle
r = The coordinates of the radius of the circle
The specified equation of a circle is; x² + y² = 9
The coordinates of the center of the circle, is therefore, O = (0, 0)
a) The coordinates of the points P and O indicates that the gradient of OP, obtained using the slope formula is; ((3·√3)/4 - 0)/(3/2 - 0) = ((3·√3)/4)/(3/2)
((3·√3)/4)/(3/2) = (√3)/2
The specified form of the gradient is; (√3)/a, therefore;
(√3)/a = (√3)/2
a = 2
The value of a is 2
b) The gradient of the tangent to a line that has a gradient of m is -1/m
The gradient of OP is; (√3)/2, therefore, the gradient of the tangent at P is -2/(√3)
The form of the gradient of the tangent at P is b/(√3), therefore;
-2/(√3) = b/(√3)
b = -2
The value of b is; -2
c) The coordinate of the point on the tangent, (0, (7·√3)/c) indicates
Slope of the tangent = -2/(√3)
((7·√3)/c - ((3·√3)/4))/(0 - (3/2)) = -2/(√3)
((7·√3)/c - ((3·√3)/4)) = (3/2) × 2/(√3) = √3
(7·√3)/c = √3 + ((3·√3)/4) = 7·√3/4
Therefore; c = 4
Learn more on the equation of a circle here: https://brainly.com/question/12182614
#SPJ1
The angle of elevation from the tip of the shadow to the top of the tree is 36°. Find the distance from the top of the tree to the tip of the shadow. Round the answer to the nearest tenth. A. 26. 7 feet b. 30. 2 feet c. 40. 8 feet d. 42. 9 feet.
Answers
The expression that represents the required distance is \(\mathbf{ s = \frac{h}{0.5878}}\)
The given parameters are:
\(\mathbf{\theta =36^o}\) --- the angle of elevation to the top of the tree
To do this, we make use of the following representations
s represents the distance from the tip of the shadow to the top of the treeh represents the height of the treeThe distance (s) is then calculated using the following sine ratio
\(\mathbf{ sin(\theta) = \frac{h}{s}}\)
Make s the subject
\(\mathbf{ s = \frac{h}{sin(\theta)}}\)
Substitute \(\mathbf{\theta =36^o}\)
\(\mathbf{ s = \frac{h}{sin(36)}}\)
Evaluate sin(36)
\(\mathbf{ s = \frac{h}{0.5878}}\)
The height of the tree is not known.
Hence, the expression that represents the required distance is \(\mathbf{ s = \frac{h}{0.5878}}\)
Read more about elevations and distance at:
https://brainly.com/question/14550522
The expression that represents the required distance is
The given parameters are:
--- the angle of elevation to the top of the tree
To do this, we make use of the following representations
s represents the distance from the tip of the shadow to the top of the tree
h represents the height of the tree
The distance (s) is then calculated using the following sine ratio
Make s the subject
Substitute
Evaluate sin(36)
The height of the tree is not known.
Hence, the expression that represents the required distance is
Read more about elevations and distances at:
brainly.com/question/14550522
#SPJ4
Wich of the following equations represents the same number?
f(x)-x-2
f(x)x+2
f(x)=-x+2
Answers
Answer:
f(x)x+2
Step-by-step explanation:
because quadrant 1 which is the y and quadrant 4 which is x are positive, therefore. f(x) x+2.
question on the image
Answers
Answer:
475.2
Step-by-step explanation:
682 is 1.55 of the original amount so 1.55x=682
682/1.55=440
440*1.08=475.2
Round each number to the nearest whole number to estimate the product of 19.875 and
4 ⅞
Answers
Rounding each number to the nearest whole number and finding there product is a s follows:
19.875 ≈ 20
\(4\frac{7}{8}\) = \(\frac{39}{8}\) = 4.875 ≈ 5
Then 20 × 5 = 100
What are whole numbers:Whole numbers are part of a real numbers that do not include fractions and decimals.
Let's convert the individual numbers to whole numbers before finding there products.
19.875 ≈ 20
\(4\frac{7}{8}\) = \(\frac{39}{8}\) = 4.875 ≈ 5
Therefore, the product of 20 and 5 are as follows:
20 × 5 = 100learn more on whole numbers: https://brainly.com/question/21109045?referrer=searchResults
Featured strategy: using a model. If four fair coins are tossed together, what are the possible head and tail configurations (HHHH, HTHH, etc.) you can obtain? Part 1 out of 3 a. Understanding the Problem. Suppose you toss two coins, coin 1 and coin 2. How many way(s) can you toss all heads? One head and one tail? Two tails? You can toss all heads way. You can toss one head and one tail ways. You can toss two tails way. There are integers from 1 to 100 that are divisible by the factors of 48 greater than 1.
Answers
There are two integers that are divisible by 48 and two that are divisible by 36. Therefore, there are three integers less than 100 that are divisible by either 48 or 36.
A model is an excellent feature for visualizing and understanding data. The possible head and tail configurations if four fair coins are tossed together are: HHHH, HTHH, THHH, HTTH, THTH, TTHH, TTTH, TTTT. In part 1, we toss two coins, coin 1 and coin 2 to understand the problem. We are interested in the number of ways we can toss all heads, one head, and one tail, and two tails. We can obtain one head and one tail in two ways. Two tails can be obtained in one way. One head can be obtained in two ways. Thus, we have three ways in which coins 1 and 2 can be tossed.Part 2 out of 3: Inclusion and Exclusion Principle. Let's examine the integers from 1 to 100 that are divisible by the factors of 48 greater than 1. We first determine the number of positive integers less than 100 that are multiples of 48. To find the number of multiples of 48 less than 100, we divide 100 by 48 to get 2, and we multiply 2 by 48 to get 96, which is the greatest multiple of 48 that is less than 100. Thus, the positive integers that are divisible by 48 and are less than 100 are 48 and 96. We now compute the number of positive integers less than 100 that are divisible by both 48 and 36. By dividing 100 by 144, we get 0, which means there are no multiples of 144 less than 100. Therefore, there are no integers from 1 to 100 that are divisible by both 48 and 36. Finally, we apply the inclusion and exclusion principle to calculate the number of integers between 1 and 100 that are divisible by either 48 or 36. The number of positive integers less than 100 that are divisible by 48 or 36 is the sum of the numbers that are divisible by 48 plus the numbers that are divisible by 36 minus the numbers that are divisible by both 48 and 36.
To know more about integers, visit:
https://brainly.com/question/490943
#SPJ11
please please help i dont understand this
Answers
Answer:
15.0
Step-by-step explanation:
answer attached
hope this helps!
I
5) From a sector of a circle subtending an angle of 135º and radius 14cm, a cup in form of a cone is made
Use it
Determine:
a) the radius of the conical cup.
(3 marks)
b) the volume of the cup
Answers
Answer:
a. 5.25cm
b. 374.6 cm^2
Step-by-step explanation:
The radius of the conical cup is half the length of the arc formed
Mathematically, the length of the arc would be;
L = theta/360 * 2 * pi * r
L = 135/360 * 2 * 22/7 * 14 = 33 cm
Now, the arc of this sector will become the base circumference of the circular base of the cone
Thus;
2 * pi * R = 33
2 * 22/7 * R = 33
R = (7 * 33)/44 = 5.25 cm
b. The volume of the cone
Kindly note that the radius of the circle becomes the height of the cone
Mathematically;
V = 1/3 * pi * r^2 * h
Kindly note that the radius of the circle will become the slang height of the cone
We can use pythagoras’ theorem to find the height of the cone
Pythagoras theorem states that the square of the hypotenuse equals the sum of the squares of the two other sides
So here, slant height l is the hypotenuse, h and r are the other two sides
So 14^2 -5.25^2 = h^2
h^2 = 168.4375
h = 12.98 cm
Volume is thus ;
1/3 * pi * 5.25^2 * 12.98 = 374.6 cm^3
If m∠4 = (5x+4)° and m∠6 = (7x-36)°, then x = ______.
Answers
Answer:
x=20
Step-by-step explanation:
The equation 15 + 10x = 7.5x + 25 represents real-world problem. Which problem best corresponds to the equation?
Answers
Step-by-step explanation:
Let's solve your equation step-by-step.
15+10x=75x+25
Step 1: Simplify both sides of the equation.
10x+15=75x+25
Step 2: Subtract 75x from both sides.
10x+15−75x=75x+25−75x
−65x+15=25
Step 3: Subtract 15 from both sides.
−65x+15−15=25−15
−65x=10
Step 4: Divide both sides by -65.
−65x
−65
=
10
−65
x=
−2
13
Answer:
x=
−2
13
please help!!!!
picture below
Answers
Answer:
7 ^ 6x + 2 = 7^15
taking log on both the sides
log 7^6x + 2 = log 7^15
6x + 2 log 7 = 15 log 7
log 7
Rylan is calculating the standard deviation of a data set that has 9 values. He determines that the sum of the squared deviations is 316.
What is the standard deviation of the data set?
Round the answer to the nearest tenth.
Answers
SS = s² (N -1)
We are given
SS = 316
N = 9
Substituting
316 = s² (9 - 1)
Solving for s
s = 6.28
The standard deviation is 6.28
What is the negative exponent rule
Answers
Answer:
a^(-n) = 1/(a^n)
In words, this rule says that if you have a base "a" raised to a negative exponent "-n", you can simplify the expression by flipping the base to the reciprocal (1/a), and changing the sign of the exponent to positive "n".
For example, let's say we have the expression 2^(-3). Using the negative exponent rule, we can simplify this as follows:
2^(-3) = 1/(2^3) = 1/8
So, 2^(-3) is equivalent to 1/8.
If these numbers represent temperatures in degrees Celsius, which is the coldest?
Answers
Answer:
-3...........……..........
Please help. Will give brainliest.
Answers
Answer:
\(T = 3 + {S}^{2} \)
Step-by-step explanation:
Fill in the table and you will see the formula above gives you the total. Substitute S=10 for you answer
How derive the formula?
You have to realize that you are give 3 to start with, so that's your starting constant. Filling in the table and subtracting 3 show help you see that the square of want is left after the subtraction is the square of the step.
Let me know is you need more clarification
You are making curtains for your little sister's dollhouse. There are 24 windows on the dollhouse and each
needs 3.75 inches of fabric for curtains. How much fabric will you need?
(please help I only have a couple of minutes)
Answers
Answer:
90 inches
Step-by-step explanation:
24 x 3.75 which is the same as 24 x 3 3/4 which is the same as
24 x 15/4 = 6 x 15 = 90
Determine the y intercept (8th grade)
Answers
Step-by-step explanation:
hey to your previous question about the slope the answer was 1 over 2
when studying the relationship between test performance and length of sleep (the night before), which type of measurement is being examined?
Answers
When studying the relationship between test performance and length of sleep (the night before), measure of association type of measurement is being examined.
What is measure of association?
Measure of association in statistics any of various factors or coefficient used to quantity a relationship between two or more variables.
Measure of association are used in various fields of research but are specially common in the areas of Epidermiology and Psycology, where they frequently are used to quantity relationships between exposures and behaviours.
Here length of sleeping and test performance is behavioural based relationship. It can best studied through measure of association.
Hence, when studying the relationship between test performance and length of sleep (the night before), measure of association type of measurement is being examined.
To know more about measure of association, click on the link
https://brainly.com/question/29576201
#SPJ4
5+6-1+5=? Please and thank you
Answers
5+6 = 11
11-1= 10
10+5=15
Hope this helps
trey and his four friends equally share a 12 ounce jar of apple sauce. how much applesauce, in ounces, does each person receive?
A. 5/12
B. 2 2/5
C. 17
D. 60
Answers
Answer:
trey and his friends can share 2.4 oz of applesauce in a 12 oz jar
Step-by-step explanation:
12÷5=2.4
If trey and his four friends equally share a 12 ounce jar of apple sauce. The \(2\frac{2}{5}\) applesauce, in ounces, does each person receive
What is Division?A division is a process of splitting a specific amount into equal parts.
Given,
trey and his four friends equally share a 12 ounce jar of apple sauce
We need to find how many ounces does each of them receives.
The number of persons including with trey are five.
We need to divide 12 ounce jar of apple sauce among five.
We just need to divide 12 by 5.
12/5
In mixed fraction we can write as \(2\frac{2}{5}\)
Hence \(2\frac{2}{5}\) ounces of apple juice , does each person receive.
To learn more on Division click:
https://brainly.com/question/21416852
#SPJ2
help me with this ques. (need solution and answer both) thanks so much <3
Answers
Answer:
1.) 27a^11 + 18a^9 -72a^7
2.) 6p^4q^3 - 10p^3q + 4p^2q3
Step-by-step explanation:
1.) Distribute and do the math
-9a^5 x (-3a^6) - 9a^5 X (-2a^4) - 9a^5 x 8a^2
27a^11 + 18a^9 -72a^7
2.) Distribute and do the math
2pq^2q x 3p^2q^2 - sp^2q x 5p + 2p^2q x 2q^2
6p^4q^3 - 10p^3q + 4p^2q3
A circle is circumscribed within a square with sides of 20 feet, as shown
above. What is the area of the circle to the nearest square foot?
Answers
On solving the provided question, we can say that - Area enclosed by the square and the circle= 400 - 314 = 86 ft sq
What is circle?Every point in the plane that is a certain distance away from a certain point forms a circle (center). It is, thus, a curve formed by points moving in the plane at a fixed distance from a point. A circle is a closed two-dimensional object where every pair of points in the plane are equally spaced out from the "center." A line that goes through the circle creates a specular symmetry line. At every angle, it is also rotationally symmetric about the center.
Side of the square =20feet
Therefore,
Radius of the inscribed circle= 20/2 = 10 feet
Area of square=20 X 20 = 400 feet sq.
Area of circle=\(\pi * r^{2} \\\pi * 10*10\\3.14*100\\314 ft sq\)
Area enclosed by the square and the circle= 400 - 314
= 86 ft sq
To know more about circle visit:
https://brainly.com/question/29142813
#SPj1
In the right triangle with 30' acute angle the long leg is 12 ft. Find other sides and the area of the triangle
Answers
We have the following:
In this case the first thing is to calculate the value of the hypotenuse, by means of the cosine function.
\(\begin{gathered} \cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}} \\ \theta=30 \\ \text{adjacent = 12} \end{gathered}\)replacing:
\(\begin{gathered} \cos 30=\frac{12}{\text{hypotenuse}} \\ \text{hypotenuse}=12\cdot\cos 30=13.86 \end{gathered}\)now, to calculate the side we use the Pythagorean theorem, as follows
\(\begin{gathered} c^2=a^2+b^2 \\ 13.86^2=12^2+b^2 \\ b^2=13.86-12^2 \\ b=6.94 \end{gathered}\)Therefore, the area is:
\(\begin{gathered} A=\frac{12\cdot6.94}{2} \\ A=41.64 \end{gathered}\)The area is 41.64 ft^2
Chelsea went to Starbucks to get coffee for her and her friends she bought two pumpkin spice lattes for $4.50 each an iced vanilla latte for $4.20 and a cake pop for $2.80. If she wants to leave a 10% tip for the barista how much will she pay in total?
Answers
Because—— she first bought 2 pumpkin slice lattes for 4.50 EACH so that would be 4.50 time 2, then you would add another 4.20. And finally add another 2.80. After you add all those numbers you will have $16. But you need to find a 10% tip. So divide the total (16 in this case) by 100. After doing this you will have 1% of your total. But you need 10% so you will get that 1 percent (0.16) and multiply it by 10. You will then get $1.6. So you now will add $16 and
In two years, the value of a car changed from $14,000 to $11,200.What is the percent of change
Answers
Answer:
Percentage change = (14000-11200) / 11200 * 100
= 280000 / 11200
= 25%
So, your final answer is 25%
Hope this helps!
Step-by-step explanation: