Find the general solution of the nonhomogeneous differential equation, 2y"' + y" + 2y' + y = 2t² + 3.

the general solution to the Cauchy-Euler equation x³y'" - 2x²y" - 2xy' + 8y = 0 is given by y(x) = c₁x² + c₂x⁻¹ + c₃x⁻¹ln(x) where c₁, c₂, and c₃ are constants.

To solve the Cauchy-Euler equation x³y'" - 2x²y" - 2xy' + 8y = 0, we'll make the substitution y = [tex]x^r[/tex], where r is a constant.

Let's differentiate y with respect to x:

y' = [tex]rx^{r-1}[/tex]

y" = [tex]r(r-1)x^{r-2}[/tex]

y'" = [tex]r(r-1)(r-2)x^{r-3}[/tex]

Now, substitute these derivatives into the original equation:

[tex]x^3(r(r-1)(r-2)x^{r-3} - 2x^2(r(r-1)x^{r-2}) - 2x(rx^{r-1}) + 8x^r = 0[/tex]

Simplifying, we get:

[tex]r(r-1)(r-2)x^r - 2r(r-1)x^r - 2rx^r + 8x^r = 0[/tex]

Combining like terms, we have:

r(r-1)(r-2) - 2r(r-1) - 2r + 8 = 0

Simplifying further, we get:

r³ - 3r² + 2r - 2r² + 2r + 8 - 2r + 8 = 0

r³ - 3r² + 8 = 0

To solve this cubic equation, we can try to find a rational root using the Rational Root Theorem or use numerical methods to approximate the roots.

By inspection, we find that r = 2 is a root of the equation. This means (r - 2) is a factor of the equation.

Using long division or synthetic division, we can divide r^3 - 3r^2 + 8 by (r - 2):

  2  |   1    -3    0    8

      |       2   -2   -4

_______________________

       1    -1   -2    4

The quotient is r² - r - 2.

Factoring r² - r - 2, we get:

r² - r - 2 = (r - 2)(r + 1)

So the roots of the equation r³ - 3r² + 8 = 0 are: r = 2, r = -1 (repeated root).

Therefore, the general solution to the Cauchy-Euler equation x³y'" - 2x²y" - 2xy' + 8y = 0 is given by:

y(x) = c₁x² + c₂x⁻¹ + c₃x⁻¹ln(x)

where c₁, c₂, and c₃ are constants.

Learn more about Cauchy-Euler equation here

https://brainly.com/question/31419735

#SPJ4

Given question is incomplete, the complete question is below

Solve the Cauchy-Euler equation:

x³y'" - 2x²y" - 2xy' + 8y = 0

Answer:

m [tex]\leq[/tex] 24

Step-by-step explanation:

-4m ≥ 96

(divide both sides by -4)

m [tex]\leq[/tex] 24

(sign flips because dividing by a negative)

Answer:

B-16

Step-by-step explanation:

jsjjsjdjsjsjxjsjsijsjdjdjjsjzjzijzjsndjidjsjsid

Let's assume the height of the TV set is h inches.

a. The height of the TV set is h inches.

Given that the length is 21 inches more than the height, the length can be represented as h + 21 inches.

b. The length of the TV set is h + 21 inches.

According to the given information, the diagonal of the TV set is 39 inches. We can use the Pythagorean theorem to relate the height, length, and diagonal:

(diagonal)^2 = (height)^2 + (length)^2

Substituting the values, we have:

39^2 = h^2 + (h + 21)^2

Expanding and simplifying:

1521 = h^2 + h^2 + 42h + 441

2h^2 + 42h + 441 - 1521 = 0

2h^2 + 42h - 1080 = 0

Dividing the equation by 2 to simplify:

h^2 + 21h - 540 = 0

We can solve this quadratic equation by factoring or using the quadratic formula. Factoring gives us:

(h - 15)(h + 36) = 0

So h = 15 or h = -36.

Since the height of the TV set cannot be negative, we discard h = -36.

Therefore, the height of the TV set is 15 inches.

Substituting this value back into the length equation, we have:

Length = h + 21 = 15 + 21 = 36 inches.

So, the dimensions of the TV set are:

a. The height of the TV set is 15 inches.

b. The length of the TV set is 36 inches.

Learn more about height of the TV set from

https://brainly.com/question/29205009

#SPJ11

f(z) = z^2 + 1 has an antiderivative on the domain G = C \ {i, -i}.

(b) Hence, we cannot determine whether f(z) = z^2 + 1 has an antiderivative on the domain G = {z in C | Re(z) > 0} based on the Cauchy-Goursat theorem alone.

(a) To determine whether f(z) = z^2 + 1 has an antiderivative on the domain G = C \ {i, -i}, we can check if f(z) satisfies the Cauchy-Riemann equations on G.

The Cauchy-Riemann equations state that for a function f(z) = u(x, y) + iv(x, y) to have a derivative at a point, its real and imaginary parts must satisfy the partial derivative conditions:

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.

For f(z) = z^2 + 1, we have u(x, y) = x^2 - y^2 + 1 and v(x, y) = 2xy.

Calculating the partial derivatives, we find:

∂u/∂x = 2x, ∂v/∂y = 2x,

∂u/∂y = -2y, ∂v/∂x = 2y.

Since ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x hold for all (x, y) in the domain G, f(z) satisfies the Cauchy-Riemann equations on G. Hence, f(z) has an antiderivative on G = C \ {i, -i}.

(b) Now, let's consider the domain G = {z in C | Re(z) > 0}. To determine if f(z) = z^2 + 1 has an antiderivative on G, we can utilize the Cauchy-Goursat theorem, which states that a function has an antiderivative on a simply connected domain if and only if its line integral around every closed curve in the domain is zero.

For f(z) = z^2 + 1, we can calculate its line integral over a closed curve C in G. However, since G is not simply connected (it has a "hole" at Re(z) = 0), the Cauchy-Goursat theorem does not apply, and we cannot conclude whether f(z) has an antiderivative on G based on this theorem.

To provide a definitive answer, further analysis or techniques such as the residue theorem may be required.

Know more about Analysis here:

https://brainly.com/question/32375844

#SPJ11

:] i honestly have no clue my self

Given that, height of the right-angled triangle = 4m. Width of the right-angled triangle = 1m. Let's assume that the rectangular enclosure will be built at the base of the right-angled triangle. The area of the rectangular enclosure can be obtained using the formula, Area of rectangle = length × breadth. Length of the rectangle = height of the right-angled triangle = 4mLet the breadth of the rectangle be x, then the length is the width of the right-angled triangle + x = 1 + x Hence, the area of the rectangular enclosure is given by: Area = Length × Breadth= (1+x)×4= 4x + 4m²Now, the maximum area can be obtained by differentiating the above expression with respect to x and equating it to zero: dA/dx = 4 = 0x = -1. This is not a valid solution since we cannot have a negative breadth, hence we conclude that the area is maximum when the breadth of the rectangular enclosure is equal to the width of the right-angled triangle, i.e., when x = 1m. Thus, Area of the rectangular enclosure = Length × Breadth= (1+1)×4= 8 m². Hence, the largest the area of Gottfried’s enclosure can be is 8 m².

To know more about Gottfried’s enclosure, click here:

https://brainly.com/question/29293732

#SPJ11

Answer:

I don't know thank answer sorry I just really need points you can report me if you want but I REALLY need some

The equation of the asymptote for the given function f(x) = 0.7(4x-3) - 2 is y = 2.8x - 4.1.

The equation of an asymptote for a function can be determined by analyzing the behavior of the function as x approaches positive or negative infinity.

For the given function f(x) = 0.7(4x-3) - 2, let's simplify it:

f(x) = 2.8x - 2.1 - 2

f(x) = 2.8x - 4.1

As x approaches positive or negative infinity, the term 2.8x dominates the function. Therefore, the equation of the asymptote can be determined by considering the behavior of the linear term.

The coefficient of x is 2.8, so the slope of the asymptote is 2.8. The y-intercept of the asymptote can be found by setting x to 0 in the equation, resulting in -4.1. Therefore, the equation of the asymptote is y = 2.8x - 4.1.

In conclusion, the equation of the asymptote for the given function f(x) = 0.7(4x-3) - 2 is y = 2.8x - 4.1.

Know more about Asymptote here:

https://brainly.com/question/32503997

#SPJ11

Answer: you know it’s 9mins but Im not sure how I should make the equation probably I think (53= 9T + 4)

Step-by-step explanation:

The correlation coefficient (p(Z, W)) between Z and W is sqrt(2) / 2.

To find the covariance of Z and W and the correlation coefficient (p(Z, W)), we can use the properties of covariance and correlation for independent random variables.

Given that X and Y are independent N(0, 2) random variables, we know that their means are zero and variances are 2 each.

Covariance:

Cov(Z, W) = Cov(7 + X + Y, 1 + Y)

Since X and Y are independent, the covariance between them is zero:

Cov(X, Y) = 0

Using the properties of covariance, we have:

Cov(Z, W) = Cov(7 + X + Y, 1 + Y)

= Cov(X, Y) + Cov(Y, Y)

= Cov(X, Y) + Var(Y)

Since Cov(X, Y) = 0 and Var(Y) = 2, we can substitute these values:

Cov(Z, W) = 0 + 2

= 2

Therefore, the covariance of Z and W is 2.

Correlation Coefficient:

p(Z, W) = Cov(Z, W) / (sqrt(Var(Z)) * sqrt(Var(W)))

To calculate p(Z, W), we need to find Var(Z) and Var(W):

Var(Z) = Var(7 + X + Y)

= Var(X) + Var(Y) (since X and Y are independent)

= 2 + 2 (since Var(X) = Var(Y) = 2)

= 4

Var(W) = Var(1 + Y)

= Var(Y) (since 1 is a constant and does not affect variance)

= 2

Now we can calculate p(Z, W):

p(Z, W) = Cov(Z, W) / (sqrt(Var(Z)) * sqrt(Var(W)))

= 2 / (sqrt(4) * sqrt(2))

= 2 / (2 * sqrt(2))

= 1 / sqrt(2)

= sqrt(2) / 2

Therefore, the correlation coefficient (p(Z, W)) between Z and W is sqrt(2) / 2.

Know more about the correlation coefficient click here:

https://brainly.com/question/29704223

#SPJ11

Answer:

[tex] m\angle SYX=76\degree [/tex]

Step-by-step explanation:

[tex] m\angle SYX = m\angle UYV[/tex]

(Vertical angles)

[tex] \because m\angle UYV=76\degree [/tex]

[tex] \therefore m\angle SYX=76\degree [/tex]

To sketch the graph of a function that has a local maximum at 6 and is differentiable at 6, we can consider a function that approaches a maximum value at 6 and has a smooth, continuous curve around that point.

In the graph, we can depict a curve that gradually increases as we move towards x = 6 from the left side. At x = 6, the graph reaches a peak, representing the local maximum. From there, the curve starts to decrease as we move towards larger x-values.

The important aspect to note is that the function should be differentiable at x = 6, meaning the slope of the curve should exist at that point. This implies that there should be no sharp corners or vertical tangents at x = 6, indicating a smooth and continuous transition in the graph.

By incorporating these characteristics into the graph, we can represent a function with a local maximum at 6 and differentiability at that point.

To learn more about differentiable click here:

brainly.com/question/24898810

#SPJ11

The solution to the Cauchy problem is x(t) = e^t and y(t) = te^t.

The Cauchy problem can be solved by finding the solution to the given system of differential equations.

In more detail, we have the following system of differential equations:

dx/dt = x - y

dy/dt = x + y

To solve this system, we can use the method of separation of variables. Starting with the first equation, we separate the variables:

dx/(x - y) = dt

Integrating both sides, we have:

ln|x - y| = t + C1

Exponentiating both sides, we get:

|x - y| = e^(t + C1)

Taking the absolute value, we have two cases:

(x - y) = e^(t + C1)

(x - y) = -e^(t + C1)

Simplifying, we obtain:

x - y = Ce^t, where C = e^(C1)

x - y = -Ce^t, where C = -e^(C1)

Next, we consider the second equation of the system. We differentiate both sides:

dy/dt = x + y

Substituting the expressions for x - y from the first equation, we have:

dy/dt = (Ce^t) + y

This is a linear first-order ordinary differential equation. We can solve it using an integrating factor. The integrating factor is e^t, so we multiply both sides by e^t:

e^t(dy/dt) - e^ty = Ce^t

We recognize the left side as the derivative of (ye^t) with respect to t:

d(ye^t)/dt = Ce^t

Integrating both sides, we have:

ye^t = Ce^t + C2

Simplifying, we obtain:

y = Ce^t + C2e^(-t), where C2 is the constant of integration

Using the initial conditions x(0) = 1 and y(0) = 0, we can find the values of the constants C and C2:

1 - 0 = C + C2

C = 1 - C2

Substituting this back into the equation for y, we have:

y = (1 - C2)e^t + C2e^(-t)

Therefore, the solution to the Cauchy problem is x(t) = e^t and y(t) = te^t.

To learn more about differential click here:

brainly.com/question/31383100

#SPJ11

x - 8 = 68

x - 8 + 8 = 68 + 8

x = 76

hope this helped

Answer:

सैम ने 9 पाउंड आटा बनाया

Step-by-step explanation:

The average rate of change for the interval from x=9 to x=10 is 39366

The standard exponential equation is given as y = ab^x

From the values of the average change, you can see that it is increasing geometrically as shown;

2, 6, 18...

In order to , find the average rate of change for the interval from x=9 to x=10, we need to find the 10 term of the sequence using the nth term of the sequence;

Tn = ar^n-1

Given the following

a = 2

r = 3

n = 10

Substitute

T10 = 2(3)^10-1
T10 = 2(3)^9
T10 = 39366

Hence the  average rate of change for the interval from x=9 to x=10 is 39366

Learn more on exponential function here: https://brainly.com/question/12940982

#SPJ1

Answer:

39,366

Step-by-step explanation:

its right

The margin of error E is approximately 0.31 oz

Margin of error is known to be a statistic expressing the amount of random sampling error in a survey's results. The margin of error informs you how close your survey findings are to the actual population's overall results. It is commonly represented by E.

The formula for margin of error is as follows:

z = critical value

σ = standard deviation

n = sample size

E = margin of error

The formula is, E = zσ/ √n

Here,

Sample size n = 51; Mean = 1.72; Standard deviation σ = 0.87 oz

Level of confidence = 97%

The level of confidence that corresponds to a z-score of 1.88 is 97% (using a standard normal table or calculator).

That is, z = 1.88 (by referring to a standard normal table or calculator)

To calculate the margin of error, we need to substitute the values in the formula

E = zσ/ √n

E = (1.88) (0.87) / √51

E = 0.3081 oz (approx)

Hence, the margin of error is approximately 0.31 oz (rounding the answer to two decimal places).

Learn more about the margin of error at:

https://brainly.com/question/32260746

#SPJ11

The exact area of the surface obtained by rotating the curve y = √(1 + eˣ) about the x-axis over the interval 0 ≤ x ≤ 7, we would need to use numerical methods to approximate the value of the integral since it does not have a simple closed-form solution.

To find the exact area of the surface obtained by rotating the curve y = √(1 + eˣ) about the x-axis, we can use the formula for the surface area of a solid of revolution.

The formula for the surface area of a curve y = f(x) rotated about the x-axis over the interval [a, b] is given by:

A = 2π∫[a, b] y * sqrt(1 + (dy/dx)²) dx

In this case, the given curve is y = √(1 + eˣ) and the interval of interest is 0 ≤ x ≤ 7. To calculate the area, we need to find the derivative dy/dx and substitute it into the formula.

Let's start by finding the derivative of y = √(1 + eˣ) with respect to x. Applying the chain rule, we have:

dy/dx = (1/2)(1 + eˣ)^(-1/2) * eˣ

Now, we can substitute y and dy/dx into the surface area formula:

A = 2π∫[0, 7] √(1 + eˣ) * sqrt(1 + [(1/2)(1 + eˣ)^(-1/2) * eˣ]²) dx

Simplifying the expression inside the integral, we have:

A = 2π∫[0, 7] √(1 + eˣ) * sqrt(1 + (eˣ/2)(1 + eˣ)^(-1)) dx

Now, we need to evaluate this integral over the interval [0, 7] to find the exact area of the surface.

Unfortunately, the integral for this particular curve does not have a simple closed-form solution. Therefore, to find the exact area, we would need to rely on numerical methods, such as numerical integration techniques or computer algorithms, to approximate the value of the integral.

Using these numerical methods, we can calculate an accurate estimate of the surface area by dividing the interval [0, 7] into smaller subintervals and applying techniques like the trapezoidal rule or Simpson's rule. The more subintervals we use, the more accurate the approximation will be.

In summary, to find the exact area of the surface obtained by rotating the curve y = √(1 + eˣ) about the x-axis over the interval 0 ≤ x ≤ 7, we would need to use numerical methods to approximate the value of the integral since it does not have a simple closed-form solution.

Learn more about curve here

https://brainly.com/question/26460726

#SPJ11

Answer:

12

Step-by-step explanation:

Through 12 years  he would have painted 72 paintings and since he hasn't sold two of them he has only sold 70.

Answer:

9√2

Step-by-step explanation:

To do this, we need to use the Pythagoras' Theorem. Which is a^2+b^2=c^2

In this case, we need to solve for C. So, we do 9^2 (A) +9^2 (B), assuming a and b are the same. So we end up with 81+81=c^2. Now, we find the square root of 162. Around 13 or 9√2

Answer:

polynomial of degree 10

Step-by-step explanation:

The degree of the polynomial is the sum of the exponents, that is

[tex]w^{7}[/tex]y³ → has degree 7 + 3 = 10

Answer: 106,000 dollars

Step-by-step explanation:

She paid 36,000 dollars as income tax and 60,000 on the rent property tax and 96,000 dollars + 10,000 dollars is 106,000 dollars.

Step-by-step explanation:

0.45 kilograms. you decide the mass value by 1000.

The value of m∠C is 85°.

Given that, △ABC ∼ △EFG. Also, m∠A = 39° and m∠F = 56°. We need to find m∠C.

Let us first write down the formula for the similarity of triangles.  The two triangles are similar if their corresponding angles are congruent.

In other words, we can write: `∠A ≅ ∠E`, `∠B ≅ ∠F`, and `∠C ≅ ∠G`.

Now, in △ABC, we have: ∠A + ∠B + ∠C = 180° (Interior angle property of a triangle)

Also, in △EFG, we have: ∠E + ∠F + ∠G = 180°(Interior angle property of a triangle)

We know that ∠A ≅ ∠E and ∠B ≅ ∠F.

Substituting these values, we get:

39° + ∠B + ∠C = 180° (From △ABC)56° + ∠B + ∠G = 180° (From △EFG)

Simplifying, we get ∠B + ∠C = 141°...(Eq 1)

∠B + ∠G = 124°....  (Eq 2)

Now, let's subtract Eq 2 from Eq 1.

We get∠C − ∠G = 17°               

Substituting values from Eq 2:

∠C − 68° = 17° ∠C = 85°

Therefore, m∠C is 85°.

T o know more about congurent angle,

https://brainly.com/question/11949261

#SPJ11

Answer:

100.00-0.22 is 99.78

U have to use decimal method. don't use Normal method

Answer:

x-intercept = 2

y-intercept = 6

Step-by-step explanation:

x-intercept:  6x = 12;  x = 2

y-intercept:  2y = 12;  y = 6

Answer:

y=-3x+5.5

Step-by-step explanation:

The answer is Y=-3x+5.5 because we already know the slope which is -3. Now all you have to do is change the Y-Intercept. As you can tell the Y-Intercept is not a whole number. So you have to change it to a decimal to get the point that you want. You can put the last part of the equation as 5.5 or 5.55. It doesn't matter but 5.5 is more official. If you find any fault in my answer let me know. Thanks. Have a good day!

Answer:

B) 44

Step-by-step explanation:

70 - 25 + 9 - 2 × 10 ÷ 2

70 - 25 + 9 - 20 ÷ 2

70 - 25 + 9 - 10

45 + 9 - 10

54 - 10

44