Q3. Given the second-order linear homogeneous ordinary differential equa- tion with variable coefficients dy - 2.0 - d.c + m(m +1)y = 0, meR, d.x2 use y(x) = 3 Anxinth to obtain 70 P} (k)a02:4–2 + P
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The given second-order linear homogeneous ordinary differential equation with variable coefficients is dy - 2.0 - d.c + m(m +1)y = 0, meR, d.x2. The solution of this equation is obtained by using y(x) = 3 Anxinth. The general solution is given by y(x) = \(c1x^{(m+1)} + c2x^{-m}\), where c1 and c2 are constants.
Given differential equation is dy - 2.0 - d.c + m(m +1)y = 0The auxiliary equation of the given differential equation is given byr^2 - 2r + m(m +1) = 0Solving the above auxiliary equation, we get r = (2 ± √(4 - 4m(m + 1))) / 2r = 1 ± √(1 - m(m + 1))Thus the general solution of the given differential equation is given by (x) = c1x^(m+1) + c2x^-m where c1 and c2 are constants. Now, using y(x) = 3 Anxinth Substitute the above value of y in the given differential equation. We get d[\(c1x^{(m+1)} + c2x^{-m}] / dx - 2[c1x^{(m+1)} + c2x^{-m}\)] - \(d[c1x^{m} + c2x^{(m+1)}] / dx + m(m+1)[c1x^{(m+1)} + c2x^{-m}]\) = 0 The above equation can be simplified as \(-[(m + 1)c1x^{m} + mc2x^{(-m-1)}] + 2c1x^{(m+1)} - 2c2x^{(-m)} + [(m+1)c1x^{(m-1)} - mc2x^{(-m)}] + m(m+1)c1x^{(m+1)} + m(m+1)c2x^{(-m-1)}\) = 0 Collecting the coefficients of x in the above equation, we get2c1 - 2c2 = 0Or, c1 = c2 Substituting the value of c1 in the general solution, we gety(x) = c1[x^(m+1) + x^(-m)] Putting the value of y(x) in the given equation, we get P(k)a0 = c1[3 Ank^(m+1) + 3 A(-k)^-m]2 = 3c1(\(Ak^{(m+1)} - A(-k)^{-m}\)) Thus ,P(k)a0 = (2/3)[\(Ak^{(m+1)} - A(-k)^{-m}\)]
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The solution to the given second-order linear homogeneous ordinary differential equation, dy/dx - 2x - d^2y/dx^2 + m(m + 1)y = 0, is y(x) = 3Anx^m.
We are given the second-order linear homogeneous ordinary differential equation with variable coefficients: dy/dx - 2x - d^2y/dx^2 + m(m + 1)y = 0, where m is a real number. To solve this differential equation, we can assume a solution of the form y(x) = Anx^m, where A is a constant to be determined.
Differentiating y(x) once with respect to x, we get dy/dx = Amx^(m-1). Taking the second derivative, we have d^2y/dx^2 = Am(m-1)x^(m-2).
Substituting these derivatives and the assumed solution into the given differential equation, we have:
Amx^(m-1) - 2x - Am(m-1)x^(m-2) + m(m + 1)Anx^m = 0.
Simplifying the equation, we get:
Amx^m - 2x - Am(m-1)x^(m-2) + m(m + 1)Anx^m = 0.
Factoring out common terms, we have:
x^m [Am - Am(m-1) + m(m + 1)An] - 2x = 0.
For this equation to hold true for all x, the coefficient of x^m and the coefficient of x must both be zero.
Setting the coefficient of x^m to zero, we have:
Am - Am(m-1) + m(m + 1)An = 0.
Simplifying and solving for A, we get:
A = (m(m + 1))/[m - (m - 1)] = (m(m + 1))/1 = m(m + 1).
Now, setting the coefficient of x to zero, we have:
-2 = 0.
However, this is not possible, so we conclude that the only way for the equation to hold true is if A = 0. Therefore, the solution to the given differential equation is y(x) = 3Anx^m = 0, which implies that the trivial solution y(x) = 0 is the only solution to the equation.
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Related Questions
2x4+6+5x8=_______________
will mark brainiest
Answers
The simplification of the given expression will be 54.
What is a numerical expression?A numerical expression is algebraic information stated in the form of numbers and variables that are unknown. Information can is used to generate numerical expressions.
Given that the expression is 2 x 4 + 6 + 5 thex 8
To solve the given expression we have to first solve the multiplication operation.
2 x 4 + 6 + 5 x 8
8 + 6 + 40
Now add the terms;
8 + 6 + 40
= 54
Hence, the simplification of the given expression will be 54.
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For the drawing, provide the missing lengths. x= hypotenuse y= leg
Answers
Answer:
Above picture is the solution for your question
Step-by-step explanation:
Use relation of trignometric ratio to get answer
Answer:
\(x = \frac{7}{ \sin(45 \degree) } \\ x = {7 \sqrt{2} } \\ \)
\(y = \frac{7}{ \tan(45 \degree) } \\ y = 7\)
\( {\boxed{ x = 7 \sqrt{2} }} \\ { \boxed{y = 7}}\)
If f(x) = 2 (3x) + 1 , what is the value of f(2) ?
Answers
Answer:
13
Step-by-step explanation:
f(x)=2(3x)+1
f(2)=2(3X2)+1
f(2)=2(6)+1
f(2)=12+1
f(2)=13
The value of function f(2) is 13.
What is a function?Function is a type of relation, or rule, that maps one input to specific single output.
Given function; f(x) = 2 (3x) + 1
We need to find the value of f(2) .
f(x) = 2(3x)+1
f(2) = 2(3 * 2)+1
f(2) = 2(6)+1
f(2)=12+1
f(2)=13
Therefore, the value of function f(2) is 13.
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In ΔDEF, the measure of ∠F=90°, the measure of ∠D=12°, and DE = 13 feet. Find the length of FD to the nearest tenth of a foot.
Answers
Answer:
1
Step-by-step explanation:
Answer:
12.7
Step-by-step explanation:
Solve for y when x = 3
K = 13
Y=?
Answers
Answer:
Y=13
Step-by-step explanation:
TIME REMAINING
59:41
Chen is bringing fruit and veggies to serve at an afternoon meeting. He spends a total of $28.70 on 5 pints of cut veggies and 7 pints of cut fruit. The food cost is modeled by the equation 5 v plus 7 f equals 28.70, where v represents the cost of one pint of cut veggies and f represents the cost of one pint of cut fruit. If the cost of each pint of fruit is $2.85, what is the approximate price of a pint of veggies? (Round to the nearest cent, as needed.)
$1.75
$2.06
$2.39
$3.99
Answers
Answer:
A pint of veggies is approximately $1.75
Step-by-step explanation:
(7 × 2.85) + 5v = 28.70. 19.95 + 5v = 28.70. 5v = 28.70 - 19.95. 5v = 8.75. v = 8.75/5. v = 1.75
Hope this helps!!
Name the property of equality or congruence that justifies going from the first statement to the second statement.
3x+x+7=23
4x+7=23
Answers
Answer:
Reflexive Property
(-7,9) and (3,-11) in slope please help
Answers
rise / run
- 11 + 9 / - 7 + 3
= - 2 / - 4
= 1 / 2
15. If gasoline costs $4.02 per gallon, your tank holds 13 gallons, and your car averages 26 miles per gallon,
a) If your tank is full, how far can you drive before you run out of gas?
b) If your tank were empty, how much will it cost to fill up your tank?
Answers
Answer: a) 338 miles
b) $52.26
Step-by-step explanation: To get the amount of miles that your car can drive before running out of gas, you just multiply how many gallons you have by how many miles each gallon lasts. Then, to get the cost of all the gas, you multiply the cost of the gas by how many gallons you need.
An airplane is flying from New York City to Los Angeles. The distance a travels in miles D is related to the time in second T by the equation D equals 0. 15t. How fast is it flying? Be sure to include the unit
Answers
An airplane is flying from New York City to Los Angeles. The distance a travels in miles D is related to the time in second T by the equation D equals 0. 15t. The plane is flying at a speed of 792 feet per second.
The distance a travels in miles D is related to the time in second T by the equation D equals 0. 15t. An airplane is flying from New York City to Los Angeles.
The formula relating distance (D), time (T), and speed (S) is given by
D = ST
This means that the speed of the plane is given by:
S = \frac{D}{T}
From the given formula, we have D = 0.15T.
We need to determine the speed S of the plane.
Substituting D, we have:
S = \frac{D}{T}
= \frac{0.15T}{T}
= 0.15
Therefore, the plane is flying at a speed of 0.15 miles per second. (Remember that the distance is in miles and time is in seconds)
We know that 1 mile is equivalent to 5280 feet. Therefore, to convert miles per second to feet per second, we multiply by 5280.
Thus, the plane is flying at a speed of 792 feet per second.
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The measure of angle b is (-8x=58). If x=-4, find the measure of angle b, then classify the angle.
Answers
Answer:
The ans is
Step-by-step explanation:
-8*-4=58
so
the ans
is
26
Haley borrows 1000 dollars from her bank.She will pay the loan at a simple annual interest rate of 4% over 4 years.How much will she pay back altogether
Answers
Answer: $49.85
No explanation needed
I need help on this question please?!
Answers
Answer:
A
Step-by-step explanation:
If you plug in 0 for w, the number of weeks, you'll see that s = 80. Meaning, at week 0 he ha 80
D(x) is the price, in dollars per unit, that consumers are willing to pay for x units of an item, and S(x) is the price, in dollars per unit, that producers are willing to accept for x units. Find (a) the eqquilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
D(x)=3000−10x, S(x) = 900+25x
(a) What are the coordinates of the equilibrium point?
______(Type an ordered pair.)
(b) What is the consumer surplus at the equilibrium point?
$____ (Round to the nearest cent as needed.)
(c) What is the producer surplus at the equilibrium point?
$____ (Round to the nearest cent as needed.)
Answers
The equilibrium point is (60, 2400), the consumer surplus at the equilibrium point is $48,000, and the producer surplus at the equilibrium point is $36,000.
(a) The equilibrium point occurs when the quantity demanded by consumers equals the quantity supplied by producers. To find this point, we need to set the demand function equal to the supply function and solve for x.
Demand function: D(x) = 3000 - 10x
Supply function: S(x) = 900 + 25x
Setting D(x) equal to S(x):
3000 - 10x = 900 + 25x
Simplifying the equation:
35x = 2100
x = 60
Therefore, the equilibrium point occurs at x = 60.
(b) Consumer surplus at the equilibrium point can be found by calculating the area between the demand curve and the equilibrium price. Consumer surplus represents the difference between the price consumers are willing to pay and the actual market price.
At the equilibrium point, x = 60. Plugging this value into the demand function:
D(60) = 3000 - 10(60)
D(60) = 3000 - 600
D(60) = 2400
The equilibrium price is $2400 per unit. To find the consumer surplus, we need to calculate the area of the triangle formed between the demand curve and the equilibrium price.
Consumer surplus = (1/2) * (2400 - 900) * 60
Consumer surplus = $48,000
(c) Producer surplus at the equilibrium point represents the difference between the actual market price and the minimum price at which producers are willing to sell their goods.
To find the producer surplus, we need to calculate the area between the supply curve and the equilibrium price.
At the equilibrium point, x = 60. Plugging this value into the supply function:
S(60) = 900 + 25(60)
S(60) = 900 + 1500
S(60) = 2400
The equilibrium price is $2400 per unit. To find the producer surplus, we need to calculate the area of the triangle formed between the supply curve and the equilibrium price.
Producer surplus = (1/2) * (2400 - 900) * 60
Producer surplus = $36,000
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*****
2.) The sum of a number ye and 6 is at least 15
Answers
Answer:
Step-by-step explanation:
This statement translates into the symbolic statement
ye + 6 ≥ 15
to the nearest tenth of a kilometer, how many kilometers are in a mile?
Answers
Answer:
1.6 kilometers
Step-by-step explanation:
Umm... sorry...
I can't think of an explanation.
I memorized the answer (because I do track and field)
Well, feel free to tell me if I did anything wrong! :)
suppose the probability of having blood type a is 0.4. choose 4 people at random and let x be the number of them with blood type a. what is the chance that nobody has type blood a?
Answers
The probability of having blood type a is 0.4, and the chance that nobody has blood type A is 0.1296, or about 12.96%.
The probability of having blood type a is 0.4.
Let x be the number of them with blood type a.
Choose 4 people at random.
The number of people with blood type a can be 0, 1, 2, 3, or 4. These are the possible values of X. If there are 4 people with blood type a, then nobody does have blood type a. Otherwise if X is less than 4, then at least one person does not have blood type a. It means we want to find P(X = 0).
X is a binomial random variable with parameters n = 4 and p = 0.4.
The probability mass function of X is given by:
P(X = x) = (n C x) * (p)^x * (1-p)^(n-x)
Now, for P(X = 0), let x = 0, n = 4, and p = 0.4.
P(X = 0) = (n C x) * (p)^x * (1-p)^(n-x)
⇒ (4 C 0) * (0.4)^0 * (0.6)^(4-0)
⇒ 0.1296
Therefore, the chance that nobody has blood type a is 0.1296.
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y-2=-3(x-4) in slope intercept form
Answers
Answer:
y = -3x + 14
Step-by-step explanation:
Slope-Intercept Form: y = mx + b
Step 1: Write point-slope form
y - 2 = -3(x - 4)
Step 2: Solve into slope-intercept form
Distribute -3: y - 2 = -3x + 12Add 2 to both sides: y = -3x + 14Cleo and jandi are having a Halloween party. They have 3 1/2 pounds of candy to divide into 14 goodie bags. How much candy will be in each bag?
Answers
Answer:
1/4 pound(s) of candy in each bag
Step-by-step explanation:
3 1/2 (pounds of candy) ÷ 14 (goodie bags) = ?
To make this easier let's turn 3 1/2 into a improper fraction and flip 14/1 (14/1 to 1/14):
3 1/2 = 7/2 14/1 = 1/14
Now we multiply:
7/2 x 1/14 = 7/28 = 1/4
(When you divide fractions, you can flip the second fraction and multiply and get your answer. Remember to simplify once you get your answer!)
Hope I helped! Sorry if wrong!
Answer:
1/4 pounds
Step-by-step explanation:
What is the area of a figure formed using a rectangle with a length of 13 kilometers and a width of 7 kilometers and a triangle with a base of 14 kilometers and a height of 11 kilometers?
Answers
Answer:
Area of rectangle= 91km
Area of triangle= 77 km
Step-by-step explanation:
see image for the working
Deion is saving up to buy a new phone. He already has $95 and can save an additional $7 per week using money from his after school job. How much total money would Deion have after 6 weeks of saving? Also, write an expression that represents the amount of money Deion would have saved in w weeks.
Answers
$95 + 7 x 6= 137
The expression that represents the amount of money Deion would have saved in w weeks is 95 + 7w and the total savings after 6 weeks will be $137.
What is an expression?A statement expressing the equality of two mathematical expressions is known as an equation.
A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.
An expression is a mathematical proof of the equality of two mathematical expressions.
As per the given,
Initial fixed money = $95
Per week saving $7/week
Total money = fixed money + money in w weeks.
⇒ 95 + 7w
For 6 weeks, w = 6
⇒ 95 + 7× 6 = $137.
Hence "The expression that represents the amount of money Deion would have saved in w weeks is 95 + 7w and the total savings after 6 weeks will be $137".
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(The question is in the picture)
I would really appreciate it please and thank you !!!
Answers
Which pair of triangles is Alessandra referring to, and which criterion should she use for establishing congruence?
Answers
Answer:
△ABCtriangle and △CDAtriangle, by side-angle-side
Step-by-step explanation:
The amount of money in the account after 18 years will be $449.5.
What are congruent triangles?If two triangles have the same size and shape they are called congruent triangles. If we flip, turn or rotate one of two congruent triangles they are still congruent. If the sides of two triangles are the same then the triangles must have the same angles and therefore must be congruent.Given is a parallelogram as shown in the image.
In order to prove ABCD as a parallelogram, we can prove ΔABE and ΔCDE congruent to each other by Angle - Angle - Side congruent property as -
∠BAE = ∠DCE {Alt. int. angles}
∠ABE = ∠CDE {Alt. int. angles}
AB = CD {Given}
Therefore, she can prove ABCD as a parallelogram by proving -
ΔABE ≅ ΔCDE using AAS criteria.
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Find the orthogonal projection of
v
v
onto the subspace W spanned by the vectors
ui.
ui.
(You may assume that the vectors
ui
ui
are orthogonal.)
v=[31−2],u1=[111],u2=[1−10]
v=⎣
⎡31−2⎦
⎤,u1=⎣
⎡111⎦
⎤,u2=⎣
⎡1−10⎦
⎤
Answers
To find the orthogonal projection of vector v onto the subspace W spanned by vectors u1 and u2, we can use the formula for orthogonal projection.
The vectors u1 and u2 are assumed to be orthogonal, which simplifies the calculation. The orthogonal projection is the closest vector in W to v, and it can be found by projecting v onto each of the vectors u1 and u2 and then summing the projections.
The orthogonal projection of vector v onto the subspace W can be calculated using the formula:
projW(v) = ((v · u1) / (u1 · u1)) * u1 + ((v · u2) / (u2 · u2)) * u2
where · represents the dot product.
In this case, we have v = [3, 1, -2], u1 = [1, 1, 1], and u2 = [1, -1, 0].
First, we calculate the dot products:
v · u1 = 31 + 11 + (-2)1 = 2
v · u2 = 31 + 1*(-1) + (-2)*0 = 2
Next, we calculate the squared lengths of u1 and u2:
u1 · u1 = 11 + 11 + 11 = 3
u2 · u2 = 11 + (-1)(-1) + 00 = 2
Using these values, we can calculate the orthogonal projection:
projW(v) = (2/3) * [1, 1, 1] + (2/2) * [1, -1, 0]
= (2/3) * [1, 1, 1] + [1, -1, 0]
= [(2/3) + 1, (2/3) - 1, (2/3)*1 + 0]
= [5/3, -1/3, 2/3]
Therefore, the orthogonal projection of v onto the subspace W is [5/3, -1/3, 2/3].
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How to solve the equation 6(m-2)=12
Answers
Linear equations
6(m−2)=12Divide both sides by 6.
m -2 = 12/6Divide 12 by 6 to get 2.
m −2 = 2Add 2 to both sides.
m = 2 + 2Add 2 and 2 to get 4.
m = 4Learn at:https://brainly.com/question/15415929Learn at:https://brainly.com/question/19260315Learn at:https://brainly.com/question/15924499Michael SpymoreThe parent function of graph C below is:
A) y=b^x
B) y=|x|
C) y=sqrtx
D) y=x^3
Answers
Answer: Well, it’s obviously not the first one. But if you could scroll down, so I could see the other graphs, that would be amazing. It would also help me answer the question!
Step-by-step explanation:
Andrea has to find a third point, C, to form a triangle on the coordinate plane shown. She is told the coordinates of its reflection point, C', across the x-axis are (2, -2) What are the coordinates of point C?
Answers
Answer:
2,2 but i could be wrong
Just wanted to say he's correct, it's 2, 2
Find the equation of the line that passes through the points ( -12, 3) and (6, -9).
Answers
m=-2/3
y-3=-2/3(x-(-12))
y-3=-2/3(x+12)
y-3=-2/3x-8
y=-2/3x-5
j^2+5.10
question: if j=2 the value of the following expression is 90.
TRUE OR FALSE?
Answers
A triangle is defined by the three points: A = (7, 7) B = (2, 2), and C = (4, 8). Determine all three angles in the triangle (in radians).
Answers
The three angles in the triangle ABC are approximately 0.45 radians (A and C) and 1.37 radians (B).
To determine the three angles in the triangle ABC, we can use the law of cosines, which relates the lengths of the sides of a triangle to the cosine of the angles opposite those sides. The law of cosines states that for a triangle with sides a, b, and c, and angles A, B, and C opposite those sides:
```
a^2 = b^2 + c^2 - 2bc cos(A)
b^2 = a^2 + c^2 - 2ac cos(B)
c^2 = a^2 + b^2 - 2ab cos(C)
```
We can use these equations to solve for the three angles in the triangle ABC.
First, we need to find the lengths of the sides of the triangle. We can use the distance formula to find the lengths of the sides AB, BC, and AC:
```
AB = sqrt((7-2)^2 + (7-2)^2) = sqrt(50)
BC = sqrt((4-2)^2 + (8-2)^2) = sqrt(52)
AC = sqrt((7-4)^2 + (7-8)^2) = sqrt(10)
```
Now we can use the law of cosines to solve for the angles:
```
cos(A) = (b^2 + c^2 - a^2) / 2bc
cos(B) = (a^2 + c^2 - b^2) / 2ac
cos(C) = (a^2 + b^2 - c^2) / 2ab
```
```
cos(A) = (50 + 10 - 52) / (2 * sqrt(50) * sqrt(10)) = 0.9
cos(B) = (50 + 52 - 10) / (2 * sqrt(50) * sqrt(52)) = 0.2
cos(C) = (10 + 52 - 50) / (2 * sqrt(10) * sqrt(52)) = 0.9
```
Now we can use the inverse cosine function to find the values of A, B, and C:
```
A = acos(0.9) ≈ 0.45 radians
B = acos(0.2) ≈ 1.37 radians
C = acos(0.9) ≈ 0.45 radians
```
Therefore, the three angles in the triangle ABC are approximately 0.45 radians (A and C) and 1.37 radians (B).
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