### Proving that an equation is harmonic.

Prove that U(x,y)=e^(-x)[xsiny-ycosy] is harmonic.

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Prove that U(x,y)=e^(-x)[xsiny-ycosy] is harmonic.

If tan(A) = 1/2, find the value of tan(5A) Hint: Use De Moivre's theorem.

Find a general solution for this differential equation. Please see the attached file for the fully formatted problem. dy/dx + 2xy = xe^(-x2+2)

For this problem state the method you used and show the work required to obtain the answer. Find the general solution for this system: this is a matrix x'= 3y+z y'= x+z+2y z"= 3y+x

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A bowl is shaped like a hemisphere with radius R centimeters. An iron ball with radius R/2 centimeters is placed in the bowl and water is poured in to a depth of 2R/3 centimeters. How much water was poured in?

How do I find the surface area obtained by revolving the curve y=x-1 from x=1 to x=4 about the line x= -1?

How does the length of the curve x=t^2, y=t^3, [0,2] come out to be 3/4 + (ln2)/2?

Find the dimensions of a cylinder with a surface area of 300 cm^2 with a maximum volume.

1. 20x=y^2 2. (x-3)^2 =1/2(y+1) 3. y2+14y+4x+45=0 Find the vertex, focus, and directrix of the parabola described by the above equations.

1. 20x=y2 2. (x-3)squared =1/2(y+1) 3. y2+14y+4x+45=0 Find an equation of the parabola that satisfies the given conditions Focus F(0-4), directrix y=4 Find the vertices, the foci and the equations of the asymptotes of the hyperbola. 1.y2divided by 49 minus x2 divided by sixteen =1 2.x2-2y2=8 Find an equat

Please see the attached file for the fully formatted problems. (i) Consider the differential equation: x. = x^2 , x(0) given x(0)>0 Find the solution of x(t) of this equation in terms of x(0) and show that there is a T, which depends on x(0), such that lim x(t) = infinity t --> T- (ii) Find the solution of the

A crude-oil refinery has an underground storage tank which has a fixed volume of 'V' liters. Due to pollutants, it gets contaminated with 'P(t)' kilograms of chemical waste at time 't' which is evenly distributed throughout the tank. Oil containing a variety of pollutants with concentration of 'k' kilograms per liter enters

Find arc length of the curve defined by the following parametric system: x=cos^-1(t) (inverse cosine) y= ln t where t is less than/equal to 1, greater than/equal to (1/sqrt 2)

Find y" for x^2/a^2 - y^2/b^2 = 1

I am trying to find the interval of convergence for the attached power series (attached as a gif). I am also supposed to check the endpoints for convergence. I'm not that good with power series and the format of this power series is really throwing me off. So I am looking for the steps to find the interval of convergence (also c

Find an equation of the tangent line to the curve, Y = x^3 - 3x^2 + 5x that has the least slope. Make sure to show all of the required steps.

A billiard ball is hit and travels in a line. If s centimeters is the distance of the ball from its initial position at t seconds, then s=100t2 + 100t. If the ball hits a cushion that is 39cm from its initial position, at what velocity does it hit the cushion?

Create a proof to show that the following is true. a x (b+c) = a x b + a x c

A special window has the shape of a rectangle surrmounted by an equilateral triangle. If the perimeter is 16 feet, what dimensions will admit the most light? (hint: Area of equilateral triangle = the square root of 3/4 times x squared.)

I am stuck on how to solve the sum of the series that I have attached in a word document.

I used the product to sum identities rule since the integral involved cosines of different angles. I have attached a word document with the integral to solve and my work. I want to know if my answer is correct. If my answer is not correct, I want to know the correct answer and the steps to get it. Thanks.

Write an equation and sketch a graph of the line through the points (-4,-3) and 3,12)

We are supposed to use the definition of the Area of a Surface of Revolution to solve this problem. I have attached this formula and the answers I received in a word document. The problem: Given: y = -x^2 + 4x defined on the closed domain [0,4] Revolve the graph about the x-axis. Find the area of the surface obtained

I need to see how to find the centroid coordinates by using integrals and moments. I have attached a word document with the formulas we are supposed to use to find the centroid. Now here is the problem: Given: y = 9 - x^2, y = 2 Find the coordinates of the centroid of the above plane region. Please refer to the atta

Given: y = -x^2 + 4x defined on the closed domain [0,4] a) sketch the graph b) Revolve the graph about the x-axis. Find the area of the surface obtained.

Given: f(x)=2x, g(x)=10 a)Sketch the plane region bounded by the functions graphs and the y-axis b)Use the shell method to find the volume of the solid formed by revolving the above plane region about the y-axis. NOTE: the graph I did is attached. The answer I got was 523.599. I'm trying to check to see if I did the gr

Given: f(x)=2x, g(x)=x, x=5 a)Sketch the plane region bounded by the functions graphs b)Use the washer method to find the volume of the solid formed by revolving the above plane region about the x-axis.

Solve each of the following differential equations: ***For each problem,state the method you used and show the work required to obtain the answer.*** 1) (y-(cos^2)x)dx + cosxdy=0 2) ye^x dx= (4+e^2x)dy

The function f(x)=2x^3 - 33x^2 + 108x - 6 has two critical numbers. The smaller one equals ______ and the larger one equals______.