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# Calculus and Analysis

### Differential Equations : Population Growth

9. The rate of change of the population of a town in Pennsylvania at any time t is proportional to the population at that time. Four years ago, the population was 25,000. Now, the population is 36,000. Calculate what the population will be six years from now. A. 43,200 B. 52,500 C. 62,208 D. 77,760 E. 89,580

### Normal Modes : Second Order Simultaneous Equations

This question is concerned with finding the solutions of the second order simultaneous equations where a = 38, b = -9, c = 378, d = -79 (i) Find the particular solutions to the differential equations which satisfy the initial conditions x = -10 and y = 7 at t = 0 together with the condition at t = 0.. For this part

### Heat Equation : Moving Source - Dirac Impulse Function

Please see the attached file for the fully formatted problem. Use superposition to solve: with boundary conditions: and initial condition

### Find the Limit : L'Hopital's Rule

Please use L'hopital's Rule to find the limit of the following expression Lim x-->0 (xsin(2x))/(1-cos(x))

### Heat Diffusion Equation and Standard Heat Equation

A) Let the temperature u inside a solid sphere be a function only of radial distance r from the center and time t. Show that the equation for heat diffusion is now: {see attachment}. This is not an exercise in doing a polar coordinate transformation. First you should derive an integral form for the equation by integrating over a

Given the following table...(a) Is y a function of x? Explain your answer. (b)Is x a function of y? Explain your answer. (See attachment for full question) Hi, I am having trouble doing these problems listed below. Please show me how to solve these problems for future reference. Thank you very much. I would like for you

### The problems are from Boundary Value Problems

The problems are from Boundary Value Problems. Undergrad 400 level course. Mainly uses partial differential skills. Some problems might require using MATLAB. Please explain each step of your solutions. Thank you very much.

### The problems are from Boundary Value Problems

The problems are from Boundary Value Problems. Undergrad 400 level course. Mainly uses partial differential skills. Some problems might require using MATLAB. Please explain each step of your solutions. Thank you very much.

### Rate of Change of Height of Cone

Gravel is dumped from a conveyor belt at the rate of 30 cubic feet per minute and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and heght are always equal. How fast is the height of the pile increasing when the pile is 10 feet high? include the correct units in your answer. Let h=the heig

### Temperature Distribution Exposed to a Hot Gas Stream

See the attached file. You solution can be similar, but IT CANNOT BE IDENTICAL OR LOOK ANYTHING CLOSE TO IDENTICAL. Please see the attached file for the fully formatted problem. L .M. Chiappetta and D.R. Sobel ("Temperature distribution within a hemisphere exposed to a hot gas stream," SIAM Review 26, 1984, p. 575?577)

### Laplace transform : Solve an Initial Value Problem

Please see the attached file for the fully formatted problem. Suppose there was an IVP such as the following: where Where and how do you begin to set this problem up to be solved using the Laplace transform? The value y(4)(t) is the fourth derivative of function y(t).

### Inverse Laplace Transforms and partial fractions

I have a transform F(s) of which I need the inverse transform for. The form of the transform is not of a common form and I am having trouble reducing it to a workable form. I am looking at a problem that requires the inverse laplace transform of f(t) to be found using the following transform: F(s) = (s*e^(-s/2))/(s^2 + p

### Laplace Equation Inside Quarter-Circle/Disk

Solve the Laplace equation inside the quarter-circle if radius {see attachment} is subject to boundary condition: {see attachment} Thank you.

### Parabolic Curve : Application to Buoyancy

You have been hired as a special consultant by u.s coast guards to evaluate some proposed new design for navigational aids buoys. The buoys are floating cans that need to be visible from some distance away without rising too far out of the water. Each buoy has a circular cross-section (viewed from below) and will be lifted with

### Rate of change of water level in a frustum of a cone

A water bucket is shaped like the frustum of a cone with height 24 inches, base radius of 6 inches and top radius of 12 inches. Water is leaking from the bucket at 10 cubic inches per minute. At what rate is the water level falling when the depth of the water in the bucket is 12 inches?

### Variation of parameters and Undetermined coefficients

1. Write a short paragraph comparing and contrasting the method of undetermined coefficients and variation of parameters. How are they similar, how they are different? If you had your choice, which method would you use? 2. Consider the differential equation: my"+cy'+ky=mg+sqrt(t) Why would the method of undetermined

### Laplace equation in cylindrical coordinates

In this problem, you will find the electrostatic potential inside an infinitely long, grounded, metal cylinder of unit radius whose axis coincides with the z-axis (See figure below). In cylindrical coordinates, the potential, V(r, theta, z), satisfies Laplace's equation... <i>Please see attached</i>... Let us assume that the po

### Linear dependency, Wronskian and Bessel's Equation

Three problems regarding the Wronskian and solutions of a second order differential equation. Example of a question 1. Determine whether the following sets of functions are linearly dependant or independent... Please see attached. 2. Bessel's equation x²y" + xy' + (x² - n²)y = 0 where n is a constant, i

### Conics, Parametric Equations, and polar Coordinates

Answers must be explained very clearly. Answers without proper justification will not be accepted. I am having a lot of trouble with these questions and the last time I posted this the TA just gave me a bunch of BS. Please take your time and answer these questions clearly and accurately with step by step work so I can follow alo

### Velocity Differential Equation

**Just need help with question 3, answers for 1 and 2 are provided*** A ping-pong ball is caught in a vertical plexiglass column in which the air flow alternates sinusoidally with a period of 60 seconds. The air flow starts with a maximum upward flow at the rate of 7m/s and at t=30 seconds the flow has a minimum (upward) flow

### Critically damped harmonic motion

This problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds (mass = 1/8 slugs) is suspended from a spring. This stretches the spring 1/8 feet. The ball is started in motion from the equilibrium position with a downward velocity of 8 feet per second. The air resistance (in pounds) of the

### Harmonic motion

A hollow steel ball weighing 4 pounds (mass = 1/8 slugs) is suspended from a spring. This stretches the spring 1/6 feet. The ball is started in motion from the equilibrium position with a downward velocity of 8 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per

### Damped harmonic motion ODE

This problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds (mass = 1/8 slugs) is suspended from a spring. This stretches the spring 1/8 feet. The ball is started in motion from the equilibrium position with a downward velocity of 8 feet per second. The air resistance (in pounds)

### Mechanical displacement

A hollow steel ball weighing 4 pounds (mass = 1/8 slugs) is suspended from a spring. This stretches the spring 1/6 feet. The ball is started in motion from the equilibrium position with a downward velocity of 8 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet pe

### Differential Equations : Rate of Change Application Problem and Wronskian

1) Miss X would like to take out a mortgage to buy a house in Leicester. The bank will charge her interest at a fixed rate of 6.1% per year compounded continuously. Miss X is able to pay money back continuously at a rate of £6000 per year. ? Make a continuous model of her economic situation, i.e. write a differential equatio

### Solving a 2nd order differential equation

Find a particular solution to the differential equation: y'' - 2y' -15y = 450t^3

### Calculating Curl of F and Potential for various n values

Please help with the following information. a) Calculate the curl of F=r^n*(xi+yj) b) For each n for which curlF=0 , find a potential g such that F=grad(g). (Hint: look for a potential of the form g=g(r), with r=sqrt(x^2+y^2). Watch out for a certain negative value of n which the formula is different.)

### Finding Eigenvalues / Eigenfunctions for Integral Operators

Please see the attached file for the fully formatted problems. Find the eigenvalues and eigenfunctions for the integral operator , where . (Answer: ) How would we solve the same problem if ? (Answer: )

### Differential Equation and Water Leaks

Leaky bucket: A bucket in the shape of a cylinder has a small hole in its bottom, through which water is leaking. Let the height of the water in the bucket be h(t), let A be the cross sectional area of the bucket and a the area of the small hole. We want to find out how long will it take the bucket to empty (see attachment for m

### Differential Equations and Mercury Levels in a Lake

A lake contains 60 million cubic meters (2MCM) of water. Each year a nearby plant adds 8.5 grams of mercury to the lake. Each year 2MCM of lake water are replaced with mercury-free water. 1. What is the differential equation that governs the amount of mercury in the lake? 2. According to your differential equation how much m