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Calculus and Analysis

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 Expansion - Exponential Product

My problem deals with the Laplace expansion property applied to a exponential product function. I am unsure as to how the problem should be solved based on the given property formula and the Differential Equation books that I have used have been of no help. If familiar, please try to use the DiPrima notation.

Find the Inverse Laplace Transform

The following is a sample test question involving the use convolution to find the inverse Laplace transform of the below equation. I have thus so far not been able to break the initial equation down into the two separate equations F(s) and G(s). Any help would be appreciated as it has been a while since I have used partial fract

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

Dynamics and Hamiltonians: Hamilton's Equations

A dynamical system, with one degree of freedom has Hamiltonian (see attachment for equation) ? Write down the Hamilton's equations governing the motion of this system, and deduce that H remains constant during the motion. ? Solve Hamilton's equations with initial conditions (see attachment) and show that q(t) and p(t) both t

Antiderivative Integral Problems

I've included the equation in the attachment. I realize that it is something larger than w, but I don't know how to get the antiderivative of f(x)f(1/x). Whoever helps me on this, please include more than just the answer in your reply. Thank you very much.

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?

Computing an Integral Using Residues

Use residue theory to compute the integral: Use residue theory to compute the integral: I know that the singular points are 4i and -4i, and that I need to integrate from -R to R and then around the half circle from 0 to pi. Only 4i is in this half-circle, so I need to calculate the residue for 4i. This is where I get stu

Antiderivative and Volume of Rotation

Please find attached a number of questions - some ask for details using mathematical software however I only require the "normal" parts to be answered.

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

Second Order Differential Equations

See the attached file. (Fun) 3. Find the general solution using the method of undetermined coefficients: y" - 2y' - 3y = t +e^-t (More Fun) 4. Find the general solution using the method of undetermined coefficients: y" + 4y = [sin

Differential equations

Consider the initial value problem for the equation of linear pendulum L. with a=0.6, b=1.7. Write this problem as an equivalent problem for a system of first-order equations. Find (analytically) the phase trajectory of this system passing through the given point (a,b). Write down the euler method for the system from

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

Differential Equations for General and Particular Solutions

1) consider the equation (non-homogenous): <i> Please see attachment for equation. </i> ? find its general solution ? find the particular solution of this equation, satisfying the initial condition y(0)=0, y'(0)=0, y''(0)=0 2) find the general solution of the differential equation (non-homogeneous) <i> Please

LaPlace transform

Use Laplace transforms to find the solution of : y'' + 10y' +25y = 180exp(t) with y(0)=1 and y'(0)=1