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# Calculus and Analysis

### Stochastic Differential Equations, Density Functions and Random Variables

We use the notation X ~N(&#956;, &#963;2) to indicate that the density function for the continuous random variable X, fx(x), has the form .... (a) If X ~N(&#956;, &#963;2) show that..... (Hint: you will need to know how to find the density function for X ? &#956; from the density function for X). (b) If ...., and X1 and X2 a

### Sturm-liouville problem

1. (a) Find the eigenvalues and eigenfunctions of the boundary-value problem. x2y'' + xy' + &#955; y = 0, y(1) = 0, y(5) = 0. (b) Put the differential equation in self -adjoint form. (c) Give an orthogonality relation. 2. Hermite's differential equation y'' -2xy' + 2ny = , n =

### Differential Equations and Green's Functions

Consider the inhomogeneous differential equation -u' + u = r(x) on (a,b) where r(x) is a continuous function on the interval (a,b) Find the solution to this differential equation in the form u = integral from a to b (G(x,s) r(s) ds) and determine G(x,s).

### General solution of non-homogenous second order linear differential equations : d²y/dx² + y = cosx

Please find the general solution of the nonhomogenous second order linear differential equation below by following these steps: 1. Find the general solution y= C1y1 + C2y2 of the associated homogenous equation (complementary solution) 2. Find a single solution of yp of above.(particular solution). 3. Express the general so

### Find the general solution of the differential equation : 2d²y/dx² + 4dy/dx + 5y = 0

Please find the general solution of the differential equation below. It needs to be converted to standard format (s² + bs + c = 0), the roots found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. Please show all details in the solution. Thank

### Graphing and Lagrange Multipliers

The plane, 4*x - 3*y + 8*z = 5, intersects the cone, z^2 = x^2 + y^2, in an ellipse. 1. Graph the cone, the plane and the ellipse 2. Using lagrange multipliers find the highest and lowest points on the ellipse. Please show grophs, equation of the ellipse and all work.

### Differential Equations : Thermodynamics of an Ideal Gas and Small Stage Efficiency

1. For a thermodynamic process involving a perfect gas, the intial and final temperatures are related by: where is the specific heat capacity of the gas, is the change of entropy and and are the initial and final temperatures of the process. Determine the value of if and . 2. The overall efficiency of a gas turbi

### Present Value of Money and Calculus

Please answer the following questions: The Present value of money is the amount that you need to deposit now in order to have a desired amount sometime in the future . 1. What is the present value needed in order to have saved \$60,000 in five years, given an annual interest rate of 7% compounded monthly? 2. What is the

### Volume of revolution..

Consider the function x=12(y^2-y^3). Determine why revolving this curve about the x-axis has a different volume than revolving about a line y=1.

### Differential Equations : Equations of Motion, Laplace Transforms

Refer to figure 2 in attachment. a) Write the equations of motion for the mechanical system.****PLEASE SHOW YOUR FREE-BODY FOR EACH MASS****THANKS B) Take the Laplace transform of these equations, arrange them in matrix form, solve for the displacement x2(t), and find the transfer function T(s)= X1(s)/F(s) C) Using the co

### Calculus : Value that allows for continuity at a point.

The function is given by the formula f(x) = (5x3 + 16x2 + 12x + 27)/(x + 3) when x<-3 and by the formula f(x) = -5x2 - x + a when -3=< x. What value must be chosen for a in order to make this function continuous at -3?

### How does a bond's value change with the interest rate?

How does a bonds value change with the interest rate?

### Differential Equation : Solve Using Classical Method; Identify Critically Damp, Overdamped or Under-Damped

X'' + 2x' + x = 5exp(-t) + t for t greater/equal to zero; x(0)=2; x'(0)=1 and identify critically damp, over-damped or under-damped (overdamped or underdamped). thank you

### Length of a curve

Using the formula for the length of a curve y=f(x) from a to b L=&#8747;&#8730;(1 + (dy/dx)²)dx Find the length of the curve: x=(y³&#8725;6) + 1/(2y) from y=2 to y=3 Hint: 1 + (dx/dy)². is a perfect square

### Identity Functions Detailed

Prove the given identity - Please explain in detail. 14. 2 cos x - 2 cos^3x = sin x sin 2x 16. cos (x-y)/cos x cosy = 1 tan x tany 20. sec x -cos x = sin x tanx 22. sin 2x = 1/tanx +cot 2x 23. If tan x = 5/12 and sin x>0 , find sin 2x 26. If sin x = -12/13 with pie <x<3pie/2, and sec y = 13/12 with 3 pi

### Finding the Volume of Revolution Using the Shell Method

Using the shell method, find the volume of a solid generated by revolving about the y axis. The boundaries of the solid are: y=9x/&#8730;(x³+9) the x-axis the line x=3 Note: I received a solution to this problem from an OTA, but it appeared that the integration wasn't complete. I would like to see another OTA sol

### Prove the following identities..."not an identity" is an option

1. cot²x-cos²x = cos²xcot²x 2. cotx = cscx/secx 3. tanx = sq.rt. sec²x-1 4. sinx/1-cotx + cosx/ 1-tanx = cosx + sin 5. cotx-1/1-tanx = cscx/secx 6. cosxsiny = 1/2 (sin(x+y) - sin (x-y))

### Inverse Laplace Transforms and Convolution Integral

Find the inverse Laplace transform of the following: (a) 10/(s2+9) (b) 2e^-2s /(s+3) (c) .... (d) Find the inverse Laplace transform of F(s) using convolution integral where F(s) = .... Please see the attached file for the fully formatted problems.

### Limits and L'Hopital's Rule (34 Problems with High Quality Solutions)

Find the limits using L'Hopital's rule where appropriate. If there is a more elementary method, consider using it. If L'Hospital's rule does not apply explain why. 1) lim as x approaches -1 (x^2 -1) / (x + 1) 2) lim as x approaches -1 (x^9 -1) / (x^5 - 1) 3) lim as x approaches -2 (x+2) / (x^2 +3x + 2) 4) lim as x approa

### Graphing, Translation, Reflection, Domains, Limits, Derivatives

For the following graph the given functions on a computer screen, how are these graphs related? 1) Y=2^x, y=e^x, y=5^x, y=20^x 2) Y=3^x, y=10^x, y=(1/3)^x, y=(1/10)^x _______________________________________________________________________ Make a sketch of the function. 7) y=4^x-3 8) y=-2x^-x 9) y=3-e^x 13)

### Inverse function Values

Please solve each problem with a detailed solution showing each step to solve the problem. Since the symbols confuse me at times please use "baby" math to show how to get from the start to the end. I understand the book in some ways, but the more I see completed the better I can think about the rest of the problems I need to d

### Evaluating an example integral

Please show the steps necessary to solve: ∫ dx ∕ (x - 2 )√(x² - 4x + 3)

### Differential Equations : Show that the differential equation

Show that the differential equation is not exact. It can be made exact by multiplying throughout by , where m and n are integers. Find m and n and hence, or otherwise, solve the equation. Please see the attached file for the fully formatted problems.

### Differential Equations : Solve the following differential equation by as many different methods as you can.

A) Solve the following differential equation by as many different methods as you can. (See attachment for equation) b) There is a type of differential equation which will always be solvable by two different methods. What type of differential equation is it and which other method can always be used to solve it? ---

### Solutions to various differential equations

Please find the attachment for the questions. Five Linear Homogeneous Differential equations that are to be solved using Variables separable and other standard methods are given in the attachment

### Differential Equations : Separation of Variables, Solve by Substitution and Integrating Factor

1. Please see the attached file for the fully formatted problems. a) Use separation of variables to solve b) Solve the following exact differential equation c) By means of substitution y=vx solve the differential equation d) By means of the substitution for approipriate values of and , solve the d

### Functions and Rate of Change Found

Find the rate of change of the function f(x) = 2x - 9 over the interval x=-1 to x=12.

### Limit : What is the limit as x approaches 0 of (2 cos x - 2 + x^2)/x^4?

What is the limit as x approaches 0 of (2 cos x - 2 + x^2)/x^4?

### General solution to differential equation

2. Use any method for xy'=2y-3x

### Cauchy principal value, residue

Verify the integral formula with the aid of residues. 1.) Show that the p.v. of the integral of (x^2+1)/(x^4+1) from 0 to infinite = (pi)/(sqrt 2). Note: p.v.=principal value; pi is approximately 3.14; sqrt 2=square root of 2 Please show all work and explain the steps, especially how you found the zeros of the