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    Calculus and Analysis

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    Simple harmonic oscillation driven by an external force

    2. A spring with a 4-kg mass has natural length 1 m and is maintained stretched to a length of 1.3 m by a force of 24.3 N. If the spring is compressed to a length of 0.8 m and then released with zero velocity, find the position of the mass at any time t. 10. As in Exercise 9, consider a spring with mass m, spring constant k,

    Initial value problem using the method for undetermined constant

    Solve the differential equation using the method of variation of parameters. Solve the differential equation or initial-value problem using the method for undetermined constants. y''-4y=e^x cosx y(0)=1, y'(0)=2 y''+y'- 2y=x - sin 2x y(0)=1, y'(0)=0 y'' + y = cot x 0<x<pi/2 See attached file for full problem description.

    Finding the Equation of a Tangent Line

    Let h(t) = ln(t) +1. Determine the equation of the line that is tangent to the graph of h(t) at the point where t =3. Provide a graph of your work showing the function and the tangent line you are identifying. (Note: You may round all decimal values in this problem to the nearest thousandth.)

    Differential Equations : Solving Boundary-Value Problems

    3. Solve the boundary-value problem, if possible. a. y''-6y'+9y =0, y(0) =1 and y(1) = 0 b. 9y''-18y'+10y = 0 , y(0) =0 and y(pie) = 1 4. If a, b and c are all positive constants and y(x) is a solution of the differential equation ay''+by'+cy = 0, show that lim x->infinity y(x) = 0

    LaPlace Transforms and Differential Equations : Masses and Springs

    In system below spring k1 is anchored at the left side and has a spring constant of k1, spring K2 has a spring constant of k2; the system is not subjected to friction or damping. Block M is subjected to a periodic driving force f(t) = A sin(&#969;t). Both masses are initially at rest in the equilibrium position Using L

    LaPlace Transforms and Differential Equations : Masses and Springs

    Two objects of mass M1 and M2 are attached to the opposite ends of a spring having a spring constant K; the entire apparatus is placed on a frictionless table. The spring is stretched and then released. Using Laplace transforms to solve the differential equations show that the period is: See attached file for full problem des

    Inverse LaPlace Transform two different ways

    Find the Inverse LaPlace Transform using different methods described in the attachment. To see the description of the problem in its true format, please download the attached question file.

    solve differential equation using Laplace transform

    1. Solve y''+4y'+4y=g(t) using Laplace transforms.... y(0) = 2; y'(0)=-3 Express the answer in terms of a convolution integral 2. Find the inverse of F(s) = 1/[(s^4)(s^2+1)] the answer can be left as a convolution integral 3. Solve y''''+5y''+4y=g(t) using laplace transforms.... y(0) = 1; y'(0)=y''(0)=y'''(0)=0 E

    Series Defined Recursively

    1) The terms of series are defined recursively by the equations: a1=2 a(n+1)=(5n+1)/(4n+3)an Determine whether ?" an converges or diverges 2) a) Show that ?" (from n=0 to infinity) [x^n/n!]converges for all x b) Deduce that lim ( n--> infinity)[x^n/n!] = 0 for all x.

    Newton's Law of Cooling

    Case: At 9:00 PM a coroner arrived at a hotel room of a murder victim. The temp. of the room was 70 degrees F. It was assumed that the victim had a body temp. of 98.6 degrees F (AT THE TIME OF DEATH)(not at 9:00PM). The coroner took the victim's temp. at 9:15 PM at which it was 83.6 degrees F and again at 10:00 PM at which it wa

    Applications of Cost Functions

    4. A hospital has seven large identical heat pumps. If one of the heat pumps malfunctions, it could seriously impact hospital operations. If preventive maintenance is performed weekly, it costs $1200 to perform preventive maintenance (PM) on all seven of them. If one of the heat pumps breaks down between PM inspections, the av

    Solving Differential Equations, Volumes of Solids and Taylor Series

    See attached file for full problem description. 1. First find a general solution of the differential equation dy 3y dx = . Then find a particular solution that satisfies the initial condition that y(1) = 4. 2. Solve the initial value problem dy y3 dx = , y(0) = 1. 3. Nyobia had a population of 3 million in 1985. Assum

    Differential Equations and Springs

    1. Solve the initial-value problems and graph the solutions on the same set of axes. y'' + 4y' + 2y = 0 y(0) = 5; y'(0) = 0 y'' + 4y' + 2y = 0 y(0) = 0; y'(0) = 5 2. Repeat problem 1 for the equation: y'' + 2y' + 5y = 0 y(0) = 5; y'(0) = 0 y'' + 2y' + 5y = 0 y(0) = 0; y'(0) = 5

    Solving Inequalities, Limits and Derivatives

    Please see the attached file for the fully formatted problems. Includes: Solve the following inequalities/equations expressing the solution. Show all work and box the final answer. Write an equation for the line described below. Graph the following function. What symmetries, if any, does the graph have? Find the v

    Continuously Compounding Interest

    Susie is opening an online savings account with an interest rate of 5.5%, compounded continuously. She wants the account to have a balance of $4,000 after 6 years. Assuming the interest rate stays the same, how much money must Susie deposit now in order to reach her goal? (Show formula and answer in a complete sentence)

    Harmonic Functions

    A)Let a be less than b and set M(z)=(z-ia)/(z-ib). Define the lines L1={z:F(z)=b}, L2={z:F(z)=a} and L3={z:R(z)=0}. The three lines split the complex plane into 6 regions. Determine the image of them in the complex plane. b) Let log be principal branch of the logarithm. Show that log(M(z)) is defined for all z in C with the