### Differential equations for the initial value problem

Assignment Help Civil Engineering
##### Reference no: EM13851905

1. A civil engineering has modelled the deflection behaviour of a cantilever beam using a second-order differential equation. Cantilevers are beams that are fixed at one end and may have a weight suspended from the other end. The weight causes the beam to bend in such a way that the deflection downwards, y, is related to the horizontal distance, x, from the fixed end, according to a differential equation of the form:

where k is determined by the stiffness of the beam and a is the length of the beam. (All distances are in metres.)

a. For a beam that is 1.5 m long, find y in terms of x and k.

b. When x = 1.5, y = 0.031. Find the value of k.

c. The quantity dy/dx gives the inclination of the beam to the horizontal. Find the angle of the beam to the horizontal when x = 0.75 m; that is, approximately at the mid-point of the beam.

2. A civil engineer needs to know the population growth pattern of a town for the purpose of infrastructure planning. He has noted that the rate of growth of the town's population (N) is 5% of its population at any time t years. If its population is currently N0 = 100 000 people (at t = 0), find the following items for using in the infrastructure planning process:

a. the population (Nt) at any time t in terms of initial population (N0) and t

b. the population in 20 years time.

3. A civil engineer is analysing a hemispherical water reservoir of radius "a". a. Show that the volume of water with a depth of h is given by Πh2/3(3a-h)

A certain hemispherical bowl has a radius of 10 metres.

b. The bowl is being filled with water at a rate of 30 m3 per minute. When the depth is 4 m, find the rate (in m per minute) at which the depth of the water is increasing.

c. Find, using calculus, the approximate change in the volume of the water in the bowl, when the depth of the water in the bowl increases from 4 to 4.01 m.

4. Use Euler's method to obtain an approximation to the value of y (2) using a step size of (i) h = 0.5 (ii) h = 0.1 (iii) 0.01.

dy/dx = y + x/yx

Where, y(1) = 2

Comment on the precision of your result (use a spreadsheet for you calculation and attach the calculation table with your assignment).

5. Two boats, A and B, start at a common point (0, 0) and move with velocity vectors

vA(t) = ti+ (t2 - t) j and vB(t) = -ti+ (t2 + t) j; t ?[0, 4].

Distance is measured in metres and time in seconds.

a. How far apart are they at t = 2?

b. What is the angle between their respective directions at t = 2?

c. At what time, if any, are the acceleration vectors of the two boats perpendicular to each other?

6. Identify the level curves of f(x, y) = √x2 + y2. Sketch a few of them.

7. Find the partial derivatives ∂f/∂x, ∂f/dy, ∂2f/∂2x, ∂f/∂y2, ∂2f/∂y∂x, ∂2f/∂x∂y for the function

f(x,y) = exy + xy

8. Find the volume V of the solid bounded by the three coordinate planes and the plane 3x+2y+5z = 6.

9. a. Find the two coupled first order differential equations for the following initial value problem

3x2y"-xy'+y=0; where y(1)=4 and y'(1)=2

b. Sketch a slope field diagram for the differential equation for the stated values of x

dy/dx = -x

x = -3, -2, -1, 0, 1, 2, 3

10. Use three numerical methods and n = 4 for approximating the value of the following integral.

02ex2dx

### Previous Q& A

#### What is the marginal rate of substitution

Mary consumes two goods, Diet Coke (X) and Cheeseburgers (Y). Mary’s Utility function is represented by U(X,Y) = 3XY2 . Her marginal utility function for X is MUx = 3Y2 . Her marginal utility function for MUy = 6XY.Write an expression for Mary’s budg..

#### Conduct a brand portfolio audit

Conduct a brand portfolio audit. My brand that I have choose is "Apple". This assignment will consist of a written element and presentation element

#### What is the marginal rate of substitution

Mary consumes two goods, Diet Coke (X) and Cheeseburgers (Y). Mary’s Utility function is represented by U(X,Y) = 3XY2 . Her marginal utility function for X is MUx = 3Y2 . Her marginal utility function for MUy = 6XY.Write an expression for Mary’s budg..

#### Determine the set of all restaurants

You and your friend are trying to decide where to have dinner. Restaurant #1 has an average price of \$20 per person and is 5 miles away. Restaurant #2 has an average price of \$30 but is only 1 mile away. Determine the set of all restaurants which wou..

#### What the optimal way to bracket stocks into portfolio

Suppose you have 5 stocks. In the past week they have changed in the following way: +5,­1,+2,­3,­2. For each of the value functions from the previous problem, determine what the optimal way to bracket stocks into a portfolio to maximize the value.

#### Value functions is loss averse

Determine if each of the following value functions is loss averse. (a) v(x) = 0.25x for x > 0 and v(x) = 0.5 for x ≤ 0   (b) v(x) = √x for x > 0 and v(x) = 0.5x for x ≤ 0   (c) v(x) = ex for x > 0 and v(x) = −√−x for x ≤ 0 (d) v(x) = √x for x > 0 and..

#### What is the deadweight loss resulting from this externality

Suppose that demand for a product is given by Q=1000-P. Supply of the product is given by Q=2P. There is a positive consumption externality from this product, in the amount of \$5 per unit. What quantity of the good is consumed under the private marke..

#### Utility function-how much of each good will he demand

Edmund has the utility function U(x, y) = 2xy + 1. The prices of x and y are both \$1 and Edmund has an income of \$20. How much of each good will he demand? A tax is placed on x so that it now costs Edmund \$2 while his income and the price of y stay t..

#### Utility function-demand for both goods double

Max has the utility function U(x, y) = x(y + 1). The price of x is \$2 and the price of y is \$1. Max’s Income is \$11. How much x does Max demand? How much y? If his income doubles and prices stay unchanged, will Max’s demand for both goods double?

#### Marginal product of capital is decreasing

What does it mean that marginal product of capital is decreasing? Show the marginal products are diminishing in the cobb-douglas case. Determination of steady state capital per worker consider the steady state capital per worker k*, how is k* affecte..

### Similar Q& A

#### Derive formulas to show that normal stresses in rods

Derive formulas to show that normal stresses in rods (1) and (2) are independent of the material used in rods, as long as they have the same modulus of elasticity, that is, E_1 = E_2.

#### Estimate the time of submersion in years of the plate

Apiece of corroded steel plate was found in a submerged ocean vessel. it was estimated that the original area of the plate was 62.5cm 2 and approximately 2.8k had corroded away during the submersion.

#### Calculate the steady state concentration of inorganic n into

Colette Lake has a surface area of 103 ha and a mean depth of 8 m. The pH of the water is 7.6. The lake receives surface water from a lake upstream, rainfall, and diffuse groundwater input

#### What is the concentration of particles in the air

A hi-vol clean filter weighs 18.00 g. After a 24 hour sampling period the filter is removed and weighs 18.65 g. The initial flow rate through the filter was 2.5 m3/min. When the filter was removed the flow rate through the filter was 1 m3/min. What i..

#### Determine the transformed moment of inertia for the beam

determine the transformed moment of inertia for the beam using steel as the reference material. Express your answer in in^4 to the nearest 0.1 in^4.

#### Find the change in depth and in absolute water level produce

Find the change in depth and in absolute water level produced by a) a smooth upward step of 1 ft b) a smooth donward step of 1 ft, in the channel bed c) find the maximum allowable size of thw upstream flow to be a possible as specified

#### How much must be deposited monthly between ages 24 and 66

You wish to retire at age 66 and at the end of each month thereafter for 30 years, to receive \$5000. Assume that you begin making monthly payments into an account at age 24.

#### Calculate for the water contained in the tank the volume

Water in the amount of 1 lb is contained in a rigid closed tank. Initially the water has a quality of 0.4 at a pressure of 10 lbf/in2. Through a stirrer, work is provided to the water in the tank until the pressure quadruples.

#### Calculate the final temperature of the gas for the process

The diaphragm is ruptured and fluid fills the entire volume of the vessel. Assume that the fluid is an ideal gas. After you have determined Qsub12, Wsub12, and delta U for the process, calculate the final temperature of the gas.

#### What should actually the separation distance

At what speed the rule of 25 feet separation between two cars is safe?

#### What real-world considerations absent in the problem

The mayor proposes financing the museum with a lump-sum tax of \$24 and then opening the museum free to the public. How many times would each person visit? Calculate the benefit each person would get from the musuem, measured as consumer surplus minus..

#### Determine what will be the discharge q at a depth

If sewage (assume the properties are the same as those of water) flows at a depth of 1.5ft in the pipe, what will be the discharge Q = 10.6 cubic feet/second #### Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!