6.2 EXERCISES - In Problems 1-14, find a general solution for the differential equation with x as the independent variable. 1. y" + 2y" – 8y' = 0 2. y" – 3y" – y' + 3y = 0 3. 6z" + 77" – z' – 2z = 0 4. y' + 2y" – 19y' - 20y = 0 3. y." + 3y" + 289' + 26y = 0 ' 0 6. y" - y" + 2y = 0 7. 2y" - y" – 10y' – 7y = 0 8. y' + 5y" – 13y' + 7y = 0 9. u" - 9u" + 27 u' - 27u = 0 10. y" + 3y" – 4y' - 6y = 0 11. y(4) + 4y" + 6y" + 4y' + y = 0 12. y' + 5y" + 3y' - 9y = 0 13. y(4) + 4y" + 4y = 0 14. y(4) + 2y" + 10y" + 18y' +9y = 0 [Hint: y(x) = sin 3x is a solution.] - = = =
Answers
To find a general solution for the differential equation, we need to find the roots of the characteristic equation and use them to form the solution.
1. The characteristic equation is r^2 + 2r - 8 = 0, which has roots r = -4 and r = 2. Therefore, the general solution is y(x) = c1e^(-4x) + c2e^(2x).
2. The characteristic equation is r^2 - 3r - r + 3 = 0, which has roots r = 3 and r = -1. Therefore, the general solution is y(x) = c1e^(3x) + c2e^(-x).
3. The characteristic equation is 6r^2 + 77r - r - 2 = 0, which has roots r = -7/6 and r = 1/6. Therefore, the general solution is z(x) = c1e^(-7x/6) + c2e^(x/6).
4. The characteristic equation is r + 2r^2 - 19r - 20 = 0, which has roots r = -4 and r = 5/2. Therefore, the general solution is y(x) = c1e^(-4x) + c2e^(5x/2).
5. The characteristic equation is r^2 + 3r + 289 + 26 = 0, which has roots r = -3/2 ± 5i/2. Therefore, the general solution is y(x) = c1e^(-3x/2)cos(5x/2) + c2e^(-3x/2)sin(5x/2).
6. The characteristic equation is r^2 - r + 2 = 0, which has roots r = (1 ± sqrt(7)i)/2. Therefore, the general solution is y(x) = c1e^((1+sqrt(7)i)x/2) + c2e^((1-sqrt(7)i)x/2).
7. The characteristic equation is 2r^2 - r - 10r - 7 = 0, which has roots r = -1/2 and r = 7/2. Therefore, the general solution is y(x) = c1e^(-x/2) + c2e^(7x/2).
8. The characteristic equation is r + 5r^2 - 13r + 7 = 0, which has roots r = -1 and r = -7/5. Therefore, the general solution is y(x) = c1e^(-x) + c2e^(-7x/5).
9. The characteristic equation is r^3 - 9r^2 + 27r - 27 = 0, which has a repeated root r = 3. Therefore, the general solution is u(x) = (c1 + c2x + c3x^2)e^(3x).
10. The characteristic equation is r^2 + 3r - 4r - 6 = 0, which has roots r = -2 and r = 3. Therefore, the general solution is y(x) = c1e^(-2x) + c2e^(3x).
11. The characteristic equation is r^4 + 4r^3 + 6r^2 + 4r + 1 = 0, which has a repeated root r = -1/2. Therefore, the general solution is y(x) = (c1 + c2x + c3x^2 + c4x^3)e^(-x/2).
12. The characteristic equation is r + 5r^2 + 3r - 9 = 0, which has roots r = -3 and r = 1/5. Therefore, the general solution is y(x) = c1e^(-3x) + c2e^(x/5).
13. The characteristic equation is r^4 + 4r^2 + 4 = 0, which has roots r = ±i. Therefore, the general solution is y(x) = c1cos(x) + c2sin(x) + c3xcos(x) + c4xsin(x).
14. The characteristic equation is r^4 + 2r^3 + 10r^2 + 18r + 9 = 0, which can be factored as (r^2 + r + 3)(r^2 + r + 3) = 0. Therefore, the repeated roots are r = -1/2 ± sqrt(11)i/2. Using the hint given, we can write the general solution as y(x) = (c1 + c2x)e^(-x/2)cos(sqrt(11)x/2) + (c3 + c4x)e^(-x/2)sin(sqrt(11)x/2).
This is a second-order linear homogeneous differential equation with constant coefficients. To find the general solution, we first propose a solution of the form y(x) = e^(rx). Plugging this into the equation, we get:
(r^2 + 2r - 8)e^(rx) = 0.
Since e^(rx) is never zero, we can divide both sides by e^(rx) and obtain the characteristic equation:
r^2 + 2r - 8 = 0.
Factoring this equation, we get:
(r + 4)(r - 2) = 0.
This equation has two distinct real roots, r1 = -4 and r2 = 2. Therefore, the general solution of the given differential equation is:
y(x) = C1 * e^(-4x) + C2 * e^(2x),
where C1 and C2 are arbitrary constants determined by initial conditions or boundary values, if given.
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Related Questions
Select the correct answer from each drop-down menu.A perpendicular bisector, CD, is drawn through point C on AB. If the coordinates of point A are (-3, 2) and the coordinates of point B are (7, 6), the x-intercept of CS is ___ Point ___lies on CD.
Answers
To answer the given questions you need to find the equation of the line CD.
Equation of a line (y=mx+b)
1. As the bisector CD is perpendicular to AB; if AB has a slope m, CD has a slope -1/m (opposite of the reciprocal)
Use the coordinates of points A and B to find the slope (m) of AB:
\(m=\frac{y_2-y_1}{x_2-x_1}\)\(m=\frac{6-2}{7-(-3)}=\frac{4}{7+3}=\frac{4}{10}=\frac{2}{5}\)Then, the slope of CD is:
\(-\frac{1}{m}=-\frac{1}{\frac{2}{5}}=-\frac{5}{2}\)2. To find the y-intercept (b); find a point in CD.
As CD is a bisector of AB, it passes through the midpoint of AB; use the next formula to find the midpoint:
\(\begin{gathered} M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\ \\ M=(\frac{-3+7}{2},\frac{2+6}{2}) \\ \\ M=(\frac{4}{2},\frac{8}{2}) \\ \\ M=(2,4) \end{gathered}\)The line CD passes through the point (2,4), use it and the slope (-5/2) to find the y-intercept as follows:
\(\begin{gathered} y=mx+b \\ 4=-\frac{5}{2}(2)+b \\ \\ 4=-5+b \\ 4+5=b \\ 9=b \end{gathered}\)3. Write the equation of the line CD: us the slope and y-intercept above
m=-5/2
b=9
\(y=-\frac{5}{2}x+9\)4. To find the x-intercept solve x using the equation of the line when y=0:
\(\begin{gathered} 0=-\frac{5}{2}x+9 \\ -9=-\frac{5}{2}x \\ \\ 2(-9)=-5x \\ -18=-5x \\ \frac{-18}{-5}=x \\ \\ x=\frac{18}{5} \end{gathered}\)X-intercept has the coordinates: (18/5,0)5. Use the equation of the line and evaluate it for each of the given points to find if the line passes through it:
\(\begin{gathered} (-52,141) \\ 141=-\frac{5}{2}(-52)+9 \\ 141=130+9 \\ 141\ne139 \\ \\ (-20,57) \\ 57=-\frac{5}{2}(-20)+9 \\ 57=50+9 \\ 57\ne59 \\ \\ (32,-71) \\ -71=-\frac{5}{2}(32)+9 \\ -71=-80+9 \\ -71=-71 \\ \\ (54,-128) \\ -128=-\frac{5}{2}(54)+9 \\ -128=-135+9 \\ -128\ne-126 \end{gathered}\)As you can see above the point that makes a truth math equation is (32,-71)
Point (32,-71) lies on CDAnswer:
The x-intercept of CD=18/5,0
Point lies on CD=32,-71
Step-by-step explanation:
Michael has a bag of marbles. The frequency of selecting each color is recorded in the table below.
Outcome Frequency
Green 4
Black 6
Orange 5
Based on the given frequency, determine the experimental probability of selecting an orange marble.
0.27
0.33
0.40
0.67
Answers
The probability of selecting an orange marble is 0.33.
Option B is the correct answer.
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
We have,
The number of times each marble is selected.
Green = 4
Black = 6
Orange = 5
Total number of times all marbles are selected.
= 4 + 6 + 5
= 15
Now,
The probability of selecting an orange marble.
= 5/15
= 1/3
= 0.33
Thus,
The probability of selecting an orange marble is 0.33.
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the sum of two numbers is 132 and their difference is 66 find the number
Answers
Answer:
99 and 33
Step-by-step explanation:
let x and y be the 2 numbers, with x being the larger of the 2, then
x + y = 132 → (1)
x - y = 66 → (2)
add (1) and (2) term by term to eliminate y
(x + x) + (y - y) = 132 + 66
2x + 0 = 198
2x = 198 ( divide both sides by 2 )
x = 99
substitute x = 99 into (1) and solve for y
99 + y = 132 ( subtract 99 from both sides )
y = 33
the 2 numbers are 99 and 33
Can someone explain how I answer this
Answers
2.2. South African friends of the Netherlands couple departed from Pretoria at 04h30am to spend the holiday with them. Their journey is described as follows: On their way from Polokwane they took the turn-off to the R521 route. Rest for 45 minutes at Dendron and took 15 minutes to do someshopping and till up the car's fuel tank at All days. 2.2.1. If the scale of the map is given as 1: 3 000 000 and the distance measured on the map between Beitbridge and Musina is 1,3 cm. calculate (in km) the actual distance between Beitbridge and Musina. ● (3)
Answers
The actual distance between Beitbridge and Musina is 3,900,000 kilometers.
To calculate the actual distance between Beitbridge and Musina, given the scale of the map and the measured distance on the map, we can use the concept of scale.
The scale of the map is given as 1:3,000,000, which means that 1 cm on the map represents 3,000,000 cm in real life.
The measured distance on the map between Beitbridge and Musina is 1.3 cm.
To find the actual distance, we can set up a proportion using the scale:
1 cm on the map / 3,000,000 cm in real life = 1.3 cm on the map / x km in real life.
Simplifying the proportion, we have:
1 / 3,000,000 = 1.3 / x.
Cross-multiplying, we get:
x = (1.3 * 3,000,000) / 1.
x = 3,900,000 / 1.
x = 3,900,000 km.
The actual distance between Beitbridge and Musina is 3,900,000 kilometers.
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If x-2 is one of the factors of x^4-kx^2+4, find the value of k.
Answers
Step-by-step explanation:
Given expression is x⁴-kx²+4 =0
x−2 is a factor of the given expression.
So, x=2
Substitute in the equation, we get
2⁴-k(2)²+4
⇒16+4k+4=0
⇒4k=−20
⇒k=−5
What pair of integers have a product of -24 and a sum of -10?
Answers
Answer:
-12 and 2
Step-by-step explanation:
Please help!!!!!!
The figure shows the graph of the quadratic function Find the average rate of change between points F and G and between points G and H.
Write down the greater rate of change.
Answers
Answer: 5
Step-by-step explanation:
Between F and G: \(\frac{-5-0}{-3-(-2)}=5\)
Between G and H: \(\frac{3-0}{-1-(-2)}=3\)
So, the greater rate of change is 5.
Can someone be nice enough to do this for me?
Answers
Answer:
1) -3, -2.5, -3/4, -3/8, 1/4, 3.1, 16/5, 6
Step-by-step explanation:
please helppp plesseee
Answers
Answer:
\(y = \sqrt{7^{2} + 3^{2} }\)
D. 7.6
Step-by-step explanation:
y² = 7² + 3²
y² = 49 + 9
y² = 58
y = (58)^½
y = Rationalize(7.61577310586) ~ 7.6
can u help plz i need it quick
Answers
Answer:
X=20°
Step-by-step explanation:
Angle in a straight line is 180°
=> 7x+40°=180°
=> 7x=180°-40°
=> 7x=140°
=> x=20°
Hope this helps you.
Picture attached below
Answers
Answer:
true
Step-by-step explanation:
hope this helps have an amazing day!
Answer:
T
Step-by-step explanation:
hope it help
Luke climbed from an elevation of 23,201 ft up to the summit of k2 which is at 28,251 ft. How many ft did Luke climb?
I need to explain the variable
write an equation
and solve it
Answers
The number of ft that Luke climbed, given the elevation he started from, and the summit, is 5, 050 ft
How to find the feet climbed?The number of feet that Luke climbed, can be found by the formula:
Number of feet climbed = Summit of K2 - Luke's starting elevation
Summit of K 2 = 28, 251 ft
Luke's starting elevations = 23, 201 ft
Elevation climbed by Luke was therefore:
= 28, 251 - 23, 201
= 5, 050 ft
In conclusion, the total elevation that Luke climbed from his starting point to the summit of K2 is 5, 050 ft.
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If |x+5|=, what are the possible values of x
Answers
The possible values of x that satisfy the equation |x+5| = c are x = c - 5 and x = -c - 5.
what is algebra?
Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.
Assuming you meant to write |x+5|= some value, I can give you a general method to solve equations involving absolute values.
If |a| = b, then either a = b or a = -b. Thus, to solve the equation |x+5| = c, where c is some given value, we can split it into two cases:
Case 1: x+5 = c
Solving for x, we get x = c - 5.
Case 2: -(x+5) = c
Solving for x, we get x = -c - 5.
So, the possible values of x that satisfy the equation |x+5| = c are x = c - 5 and x = -c - 5.
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Solve the equation.
9x^3 - 36x = 0
Answers
Answer:
x = -2,0,2
Step-by-step explanation:
9x^3 - 36x = 0
Factor out 9x
9x(x^2 -4) =0
Factor inside the parentheses, which is the difference of squares
(a^2 -b^2 = (a-b)(a+b)
9x (x-2)(x+2) = 0
Using the zero product property
9x =0 x-2=0 x+2 =0
x=0 x=2 x=-2
x = -2,0,2
Compute the periodic rate and interest in the first period for a $2,700 loan with 7.5% APR for the following periods. a. Monthly b. Daily (use a non-leap year) c. Quarterly (4 times a year)d. Biweekly (every 2 weeks)
Answers
The periodic rate and interest in the first period for a $2,700 loan with 7.5% APR are 0.625% and $16.88 monthly, 0.0205% and $1.65 daily, 1.875% and $50.63 quarterly, and 0.288% and $7.80 biweekly.
The periodic rate is the interest charged for one period, and the interest in the first period is the amount of interest charged in the first period.
a. Monthly:
Periodic rate = APR / number of periods in a year = 7.5% / 12 = 0.625%Interest in the first period = Loan amount x periodic rate = $2,700 x 0.625% = $16.88b. Daily:
Periodic rate = APR / number of periods in a year = 7.5% / 365 = 0.0205%Interest in the first period = Loan amount x periodic rate x number of days in the first period = $2,700 x 0.0205% x 30 = $1.65c. Quarterly:
Periodic rate = APR / number of periods in a year = 7.5% / 4 = 1.875%Interest in the first period = Loan amount x periodic rate = $2,700 x 1.875% = $50.63d. Biweekly:
Periodic rate = APR / number of periods in a year = 7.5% / 26 = 0.288%Interest in the first period = Loan amount x periodic rate = $2,700 x 0.288% = $7.80To learn more about the Interest, visit:
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The fact that two people taking the same test in two different places will receive the same instructions, the same questions, and the same time limits means that the test has been
Answers
The fact that two people taking the same test in two different places will receive the same instructions, the same questions, and the same time limits means that the test has been standardized.
Standardized tests are designed to have a uniform format, including consistent instructions, the same questions, and equal time limits to ensure that the scores are reliable, valid, and consistent across different test-takers and testing locations. The standardized testing system enables educators to assess a student's academic performance in a specific subject and to compare students' knowledge and skills. Therefore, it is easier for teachers to compare students' results to decide which students have achieved the desired level of academic performance and those who need additional support or interventions. Standardized tests play a critical role in determining a student's academic progress, educational opportunities, and career prospects. Standardized tests can be used to identify students' strengths and weaknesses, provide information about how students are learning, and to evaluate the effectiveness of educational programs.
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please help me answer this question asap
Answers
Answer:
It's quite easy
Step-by-step explanation:
people less than 30 years = frequency of people 0 to 15 + 15 to 30 = 8+15 =23
Therefore there are 23 people less than 30 years old.
pls mark me as brainliest pls.
Find the area of sphere of radius 4 cm
Answers
Step-by-Step Solution:
Radius (r) = 4 cm
Area of the Sphere = 4πr^2
Therefore,
= 4πr^2
= 4 * π * r * r
= 4 * 22/7 * 4 * 4
= 4 * 22/7 * 16
= 88/7 * 16
= 1408/7
=> 201.14
Area of the Sphere = 201.14 cm^2
Evaluate the expression for the given value.
3x(x−2), when x=7 .
Answers
Answer:
\(\huge\boxed{Answer\hookleftarrow}\)
Given,
\(x = 7\)
Substitute the value of 'x' in the equation 3x (x - 2).
One way :-
\(3x(x - 2) \\ = 3 \times 7(7 - 2) \\ = 21(5) \\ = 21 \times 5 \\ = 105\)
Second way :-
\(3x(x - 2) \\ = 3x {}^{2} - 6x \\ = 3 \times (7) {}^{2} - 6(7) \\ = (3 \times 49) - (6 \times 7) \\ = 147 - 42 \\ = 105\)
✐ The answer is 105.
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# ꧁❣ RainbowSalt2²2² ࿐
below are graphs of functions over the interval (-4,4). find the following for each function: The domain, the zeros ( the value of x where the value of the function is 0) and the intervals where the interval is positive and intervals where it is negative
Answers
From the graph, we get that:
The domain is (-4,4).The function has no zeros.The function is positive in the interval (-4, 0].The function is negative in the interval (0,4).------------------------------------
The domain of a function is the set that contains all the possible input values. In a graph, it is the values of the x-axis, that is, the horizontal axis.In this question, the values of x are in the interval of (-4,4) thus, the domain is (-4,4).The zeros are the values of x for which y = 0, that is, the values of x where the function crosses the horizontal axis. In this question, the function does not cross the horizontal axis, thus, it has no zeros.The function is positive when the graph is above the horizontal axis. In this question, it is in the interval (-4,0].The function is negative when the graph is below the horizontal axis. In this question, it is in the interval (0,4).A similar problem is given at https://brainly.com/question/24493729
Answer: No Zeros
Domain: -4<=x<=4
Positive: -4<=x<=0
Negative: 0<x<=4
Step-by-step explanation: The other person had a good explanation but the problem needed a different format.
Lines a and b are perpendicular. Which statements below must be true? Select all that apply. The slopes of lines a and b are reciprocals Line a and line b intersect at their midpoints. When the slopes of lines a and b are multiplied, the product is −1. Line a and line b intersect to form four right angles. The slopes of lines a and b are equal. The slopes of lines a and b are opposites
Answers
Answer:
When the slopes of lines a and b are multiplied, the product is −1.
Line a and line b intersect to form four right angles.
Step-by-step explanation:
Given
Perpendicular lines a and b
Required
Select the true statements
Represent the slope of line a with \(m_1\) and the slope of b with \(m_2\).
The condition for perpendicularity is:
\(m_1 * m_2 = -1\)
Solving further:
\(m_1 = -\frac{1}{m_2}\)
To solve further, we need to analyze each of the given options
(a) This is not true because the slope of perpendicular lines are not reciprocal
(b) This is also not true because perpendicular lines do not necessarily have to intersect at their midpoints
(c) This is true because of the condition stated above: \(m_1 * m_2 = -1\)
(d) This is also true because perpendicular lines form right angles when they intersect
(e) This is not true because of the condition stated above: \(m_1 = -\frac{1}{m_2}\)
(f) This is not true because of the condition stated above: \(m_1 = -\frac{1}{m_2}\)
The product of the slope of the perpendicular lines is negative of one. Then the correct options are B and C.
What is the linear system?It is a system of an equation in which the highest power of the variable is always 1. A one-dimension figure that has no width. It is a combination of infinite points side by side.
Lines a and b are perpendicular.
Then the product of the slope of the perpendicular lines is negative of one. That is given as
\(\rm m_1*m_2 = -1\)
a. The slopes of lines a and b are reciprocals Line a and line b intersect at their midpoints. This is false.
b. When the slopes of lines a and b are multiplied, the product is −1. This is true.
c. Line a and line b intersect to form four right angles. This is true.
d. The slopes of lines a and b are equal. This is false.
e. The slopes of lines a and b are opposites. This is false.
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Drag each capacity to match it to an equivalent capacity.
8L
80 mL
0.08 L
800 mL
Capacity
Equivalent Capacity
0.08 L
8,000 mL
0.8 L
80 mL
Answers
8L= 8,000 mL
80 mL= 0.08 L
0.08 L= 80 m L
800 mL = 0.8 L
Round 672,641 to the nearest ten thousand.
Answers
Answer:
673,000 i think hope this helps have a nice day please give me brainliest if correct :))
Step-by-step explanation:
Solve for x in this equation.
7^3x-17 = 49^x
Answers
Answer:
X = 17
Step-by-step explanation:
If you created equivalent expressions with equal bases that should be the answer.
Write in exponential form: 7^3x-17 = 7^2x
3x-17=2x
-2x -2x
————-
X=17
the frequency polygon and the histogram are two different ways to represent the same data set. True or false?
Answers
False. The frequency polygon and the histogram are two different ways to represent data, and they have distinct characteristics.
A frequency polygon is a line graph that displays the distribution of a quantitative variable. It shows the frequency of each value or class interval on the horizontal axis and the corresponding frequencies on the vertical axis. The points on the graph are connected to form a line, which gives a visual representation of the data distribution.
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Sierra is riding on a carousel with a 10 foot radius sheet if she's located 1.5 feet from the edge of the car so how far will she travel five revolutions
Answers
She will travel a distance of 62.8 feet * 5 revolutions = 314 feet.
To know how far Sierra will travel after five revolutions on the carousel, we will need to calculate the circumference of the carousel (the distance around the outside) and then multiply it by the number of revolutions.
The circumference of the carousel can be calculated using the formula:
Circumference = 2 * pi * radius
Where "pi" is the irrational number approximately equal to 3.14, and "radius" is the radius of the carousel (in this case, 10 feet).
Plugging in the values, we get:
Circumference = 2 * 3.14 * 10 feet
Circumference = 62.8 feet
Since Sierra will make five revolutions, she will travel a distance of 62.8 feet * 5 revolutions = 314 feet.
Keep in mind that this is just an approximation, as the actual distance traveled may be slightly different due to factors such as the speed of the carousel and Sierra's position on the carousel.
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I need this math answer now and explain it to me please
Answers
Answer:
C
Step-by-step explanation:
i dont have an explanation its just C
Answer:
the answer is 35.
Step-by-step explanation:
So first you look for 75 on the x-axis because this represents the number of people in attendance. 75 would be the line between 50 and 100. Then you need to follow the 75 line up until you see the point that the graphed line intersects the "75 line". When you drag you finger from that point over to the y-axis, you see the number of recyclables is about 35.
True or False: A shift is another type of permutation
Answers
It is true that the A shift is a type of permutation where elements in a sequence or set are moved to the right or left by a certain number of positions.
For example, shifting the sequence (1,2,3,4,5) to the right by 2 positions would result in the sequence (4,5,1,2,3). This can be represented mathematically as (1,2,3,4,5) → (4,5,1,2,3), which is a permutation of the original sequence. Shifts can be useful in various applications, such as cryptography or data compression, where rearranging elements in a sequence can improve efficiency or security.
True. A shift is a specific type of permutation. In general, a permutation refers to the arrangement of objects in a particular order. A shift, also known as a cyclic permutation, involves moving elements of a set in a fixed direction, such that the order is maintained, but the elements change positions. In a shift, the first element moves to the last position, while all other elements move one position forward. This maintains the relative order of the elements and results in a new arrangement. Therefore, a shift is a type of permutation, as it represents an altered ordering of the elements in a set.
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Adam is going to make concrete.
He is going to use:
180 kg of cement
375 kg of sand
1090 kg of stone
Cement, sand and stone are sold in bags:
1 bag cement 25 kg
1 bag sand 22.5 kg
1 bag stone 50 kg
Adam already has:
8 bags of cement
19 bags of sand
20 bags of stone
Work out what bags he needs to buy to make the concrete.
Working must be shown please.
Answers
Answer:
Stone
Step-by-step explanation:
The answer is on the paper.
Answer:
Step-by-step explanation:
