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# Calculus and Analysis

### 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

### Second order differential equation

Find particular solution to: y'' + 25y = -20sec(5t)

### Particular Solution ODE

Find a particular solution to: y'' + 6y' + 9y = (-11e^(-3t)) / (t^2 + 1).

### Euler-Cauchy Equation

Consider the differential equation: r^2*R"+r*R'-R=0 a) Find all values of n for which the function R=r^n is a solution to the differential equation. Do this by substituting {the solution into the DE and seeing which values of n will make the equation true. b) Solve the initial-value problem (IVP) with R(1)=2 and R'(1)=0.

### Linear Differential Equations

1) Consider the equation... ? Find its general solution. ? Find the particular solution of this equation satisfying the initial condition. 2) Find the general solution of the differential equation... ? Prescribe any concrete initial data for which this equation has a unique solution. ? Find the particular solution of th

### 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

### Solve Three Differential Equations

Find the general solution of the following system of differential equations: i.e. 1) dx/dt = x-2y-t2 (FOR COMPLETE PROBLEMS PLEASE SEE ATTACHMENT)

### Concepts Behind Limits

Please explain the concepts behind limits.

### 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: )

### Solution to Second Order Order differential Equation

Find solution of y'' + 6y' = 80exp((2)t)) with y(0) = 8 and y'(0)=5

### 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

### Solution to Nonlinear Differential Equation

Please see the attached file for the fully formatted problems. 1. Consider the nonlinear differential equation attached a. Find the solution to this differential equation satisfying y(0) = y0 where y0 does not equal +/- 1. What is the solution if y0 = +/-1. b. What happens to the solution as t--> infinity for y0 > -1?

### Equation of the Tangent Line

Find the equation of the tangent line to the curve y=((x-sqrt x)/(x*sin x)) at x=(5pi/6).

### First Order Differential Equation

Problem: First find the general solution of the linear ODE in each IVP by following the steps of the procedure. Then use the initial condition to find the solution of the IVP. Discuss that solution's qualitative behaviour as t --> +(SYMBOL). Give the largest t-interval on which the solution is defined: y' + 2y = 3, y(0) = 1

### First order linear non-homogenous differential equation.

Use the five steps of the Method of Integrating Factors to find the general solution of each linear ODE (hint: write ODE in normal linear form): y' - 2ty = t y' = sin(t) + y*sin(t) (PLEASE SEE ATTACHMENT FOR COMPLETE PROBLEM)

### Ode (Boundary Condition, Implicit and Explicit Solutions)

Consider the ODE y' = y2/x subject to the boundary condition y(1)=1. Find an implicit solution of the form H(x,y) = constant, then find an explicit solution of the form y=y(x). What is the largest x-interval on which the solution is defined? *(Please see attachment for proper citation of symbols and numbers)

### Differential Equations

For each of the following ordinairy differential equations, indicate its order, whether it is linear or nonlinear, and whether it is autonomous or non-autonomous. a) df/dx +f^2=0 (See attachment for all questions)

### Multiple integrals

Consider the vector field F(x,y)= (-yi+xj)/(x^2+y^2) Question1)Show that F is the gradient of the polar angle function teta(x,y)=arctan(y/x) defined over the right half-plane x>0 . Question2)Suppose that C is a smooth curve in the right half-plane x>0 joining two points : A:(x1,y1) and B(x2,y2).Express "integral(F.dr)"on

### Area of Ellipse - Multiple Integrals

Find the area of the ellipse (x+7y-11)^2 + (5x-3y+9)^2 < 1

### Integral Equation Solved

Please show me how to solve this equation - can it be solved by substitution or am I on the wrong track? *(Please see attachment for equation)

### Y as a function of x

Find y as a function of x if: (x^2)(y'') + 15xy'+49y = 0 y(1) = -4 y'(1) = -3

### Instantaneous rate of change: Pool problem

A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. If the pool is being filled at a rate of 0.8 ft^3/min, how fast is the water level rising when the depth at the deepest point is 5 ft?

### Rate of change

A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at a rate of 0.2 m^3/min, how fast is the water level rising when the water is 30 cm deep?

### Instantaneous Rate of Change of Water

A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft^3/min, how fast is the water level rising when the water is 6 inches deep?

### Calculus and Analysis: Functions

See attached for details.