Homework 1 Due Monday, September 17th at the beginning of class. Show your work. 1. Match the diﬀerential equation in (a)-(c) to a family of solutions in (d)-(f). The point of this exercise is not to solve the diﬀerential equations in a) - c). (a) y = y 2 (b) y = 1 + y 2 (c) yy = 3x (d) y = tan(x + C) (e) 3x2 − y 2 = C (f) y = −1/(x + C)

2. Find the value of k so that y = e3t + ke2t is a solution of y − 2y − 3y = 3e2t . 3. Solve the following diﬀerential equations and IVP’s. You may solve these equations implicitly. (a) y + 3x2 y = x2 (b) y ln t = y+1 t 2

dy dt

2 sin t dy = 0 y (d) y y − t = 0, y(1) = 2, y (1) = 1 (c) cos t dt + 1 + (e) y − y = et y 2 (f) cos(xy) − xy sin(xy) + 2xyex + (ex − 2y − x2 sin(xy))y = 0, y(1) = 0 dy x + 3y (g) = dx 3x + y 2 x sin x dy + (y cos3 x − 1) dx = 0, 0 < x < π (h) cos (i) (x + yey/x )dx − xey/x dy = 0, y(1) = 0 [Hint: Think homogeneous.] 4. Suppose the diﬀerential equation dP = (k cos t)P, dt is a model of the human population P (t) of a certain community, where k is a positive constant. Discuss a (non-morbid) interpretation for the solution of this equation. In other words, what kind of population do you think it describes? [Actually solving the equation is not helpful.] 5. Write down a diﬀerential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. (Remarks: Consider the forces acting on a falling object, and what they must add up to by Newton’s Second Law. You do not need to solve the diﬀerential equation.) 2 2

6. The ﬂu makes its ﬁrst appearance in Pleasant Island (population 999) when Mrs. Smith’s niece arrives for a visit not feeling well. Let y(t) be the number of infected people. Suppose that apart from the initial visitor, no one enters or leaves the island for an indeﬁnite period of time. Suppose also that the number of islanders infected grows at a rate proportionate both to the number of people infected and the number...

...algebraic expansion of powers of a binomial.
Figure 1 : Example use the binomial Expansion in geometric
There are 3 methods to expand binomial expression
Method 1 - Algebra method
Expansion two or more expression.
Example: The expansion depend on power value (n)
n = 0, (a + x)0 = 1
n = 1, (a + x)1 = a + x
n = 2, (a + x)2 = (a + x) (a + x) = a2 + 2ax + x2
n = 3, (a + x)3 = (a + x) (a + x) (a + x) = a3 + 3a2x + 3ax2 + x3
n = 4, (a + x)4 = (a + x)(a + x)(a + x)(a + x) = a4 +4a3x +6a2x2 +4ax3+ x4
Method 2 - PASCAL Triangle
Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician Blaise Pascal
Base on algebra method.
only using the coefficients of terms.
Power value Coefficient
n = 0 1
n = 1 1 1
n = 2 1 2 1
n = 3 1 3 3 1
n = 4 1 4 6 4 1
n = 5 1 5 10 10 5 1
n = 6 1 ? 1
Example:
(1 + 2x)5
n = 5 1 5 10 10 5 1
(1 + 2x)5 =
=
Method 3 - Binomial theorem
Sum of terms (Hasil tambah sebutan)
The general...

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Determination of Drag Coefficients for Various Sphere Types
Adriana Carbon, Jessica Lake, Jonathan Bessler
ChE 341 Experiment 1
March 6, 2015
Abstract
“A wind tunnel is a specially designed and protected space into which air is drawn, or blown, by mechanical means in order to achieve a specified speed and predetermined flow pattern at a given instant” . The flow over a specific object can be observed from outside the wind tunnel through transparent windows that enclose the test section and the flow characteristics can be measured through specialized instruments.
In this experiment an open-type wind tunnel was used to observe the flow over different types of spheres and the experimental results were used to calculate each sphere’s flow characteristics, such as the coefficient of drag force (CD).
From our experimental results, we determined that the balls all displayed a sharp drop in drag coefficient at the turbulent transition with drag values in reasonable ranges when comparing these results to the literature values.
Introduction:
The total drag of an object may be due to pressure as well as frictional effects. In this situation the coefficient of drag, CD, is defined as
Equation 1
where F is the force, AP is the projected area of the surface, is the density of the fluid, and v∞ is the free-stream fluid velocity. The quantity is often called the dynamic pressure. This equation can be used to...

...Understanding the Pearson Correlation Coefficient (r)
The Pearson product-moment correlation coefficient (r) assesses the degree that quantitative variables are linearly related in a sample. Each individual or case must have scores on two quantitative variables (i.e., continuous variables measured on the interval or ratio scales). The significance test for r evaluates whether there is a linear relationship between the two variables in the population. The appropriate correlation coefficient depends on the scales of measurement of the two variables being correlated.
There are two assumptions underlying the significance test associated with a Pearson correlation coefficient between two variables.
Assumption 1: The variables are bivariately normally distributed.
If the variables are bivariately normally distributed, each variable is normally distributed ignoring the other variable and each variable is normally distributed at all levels of the other variable. If the bivariate normality assumption is met, the only type of statistical relationship that can exist between two variables is a linear relationship. However, if the assumption is violated, a non-linear relationship may exist. It is important to determine if a non-linear relationship exists between two variables before describing the results using the Pearson correlation coefficient. Non-linearity can be assessed visually by examining a scatterplot of...

...Practice Problems Set – 1 MEC301: Heat Transfer
Q.1 The slab shown in the figure is embedded on five sides in insulation materials. The sixth side is exposed to an ambient temperature through a heat transfer coefficient. Heat is generated in the slab at the rate of 1.0 kW/m3. The thermal conductivity of the slab is 0.2 W/m-K. (a) Solve for the temperature distribution in the slab, noting any assumptions you must make. Be careful to clearly identify the boundary conditions. (b) Evaluate T at the front and back faces of the slab. (c) Show that your solution gives the expected heat fluxes at the back and front faces.
Q.2
Compute overall heat transfer coefficient U for the slab shown in the figure.
Given: Ls = 2 mm = 0.002 m Lc = 3 mm = 0.003 m ks = 17 W/m-K kc = 372 W/m-K Q.3 A 4 mm diameter spherical ball at 50oC is covered by a 1 mm thick plastic insulation (k = 0.13 W/m-K). The ball is exposed to a medium at 15oC, with a combined convection and radiation heat transfer coefficient of 20 W/m2-K. Determine if the plastic insulation on the ball will help or hurt heat transfer from the ball. Q.4 Prove that if k varies linearly with T in a slab, and if heat transfer is one-dimensional and steady, then q may be evaluated precisely using k evaluated at the mean temperature in the slab.
Q.5 Layers of equal thickness of spruce and pitch pine are laminated to make an insulating material. How should the laminations be oriented in...

...GINI COEFFICIENT
The Gini coefficient is a summary statistic that measures how equitably income is distributed in a population. It is derived from the Lorenz curve which plots the cumulative percentages of the income and the population against which it is distributed. The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient is the ratio of the area that lies between the line of equality and the Lorenz curve divided by the total area under the line of equality. A Gini coefficient of 0 represents perfect equality, while an index of 100 implies perfect inequality. A low Gini coefficient indicates a more equal distribution of wealth among the population while a higher one represents more inequality. The Gini co efficient for was Pakistan was around 0.68 in 2006 while Sweden’s was 0.24.The ideal Gini coefficient lies between 0.25 and 0.40 because people at different levels of specialization cannot be paid equally and there is disincentive to work. Similarly, at levels where income equality is too high law and order situation worsens affecting development negatively. By using this we cannot only find out wealth and income equality of a population but also compare the income distribution of two different sectors or countries. It also indicates how the distribution of income has changed within a country over a period of time. However, even though countries may have...

...3/16
Chem 322 Friday
Partition Coefficient (Extraction) Lab for Benzoic Acid
Reaction Mechanism:
Purpose:
The purpose of this particular lab was to demonstrate the difference in separation of Benzoic acid in water and Dichloromethane. Also the difference in separation of Benzoic acid in bicarbonate and Dichloromethane. By calculating the partition coefficient for each of these separations, it became clear the differences in the separation of the aqueous and organic layers.
Observations & Results:
In each of these separations, when the layer were mixed together and allowed to settle, two distinct layers were formed. The aqueous layer was on top and the organic layer was on the bottom. In both cases, gaseous pressure was built up in the separatory funnel but in the bicarbonate separation, the gas build up was much greater due to the reaction giving off carbon dioxide. When each product was weighed, a white crystalized structure had formed in the erlenmeyer flask.
Calculations:
Partition Coefficient for Water
K = .36g organic layer / .04 grams water layer
K = 9.0
Partition Coefficient for Bicarbonate
K = .02g organic layer / .40g bicarbonate layer
K = .05
Conclusion:
From this lab, we have learned that the partition coefficient for benzoic acid can differ greatly depending on what the aqueous layer in the separation is. For example, Benzoic acid is soluble in...

.............................................2
c) Material & Apparatus..............................................................................3
d) Procedure...............................................................................................3
e) Results & Discussion..............................................................................4
f) Conclusion..............................................................................................5
g) References..............................................................................................5
OBJECTIVE
This experiment was carried out to study the conduction of heat along a composite bar and evaluate the overall heat transfer coefficient.
INTRODUCTION
Conduction is defined as the transfer of energy from more energetic particles to adjacent less energetic particles as a result of interactions between the particles. In solids, conduction is the combined result of molecular vibrations and free electron mobility. Metals typically have high free electron mobility, which explains why they are good conductors.
Conduction can be easily understood if we imagine two blocks, one hot and the other cold. If we put these blocks in contact with one another but insulate them from the surroundings, thermal energy will be transferred from the hot block to the cold block. This mode of heat transfer between the two solid blocks is termed as ‘conduction’.
Figure...

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Coyne and Messina Articles, Part 3 Spearman Coefficient Review
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Coyne and Messina Articles, Part 3 Spearman Coefficient Review
The Spearman Correlation Coefficient remains one of the most important nonparametric measures of statistical dependence between two variables. The Spearman Correlation Coefficient facilitates the assessment of two variables using a monotonic function. This representation is only possible if the variables are perfect monotones of each other and if there are no repeated data values. This enables one to obtain a perfect Spearman correlation of either +1 or -1. The Spearman correlation coefficient nonparametric because, a perfect Spearman correlation results when X and Y are related by any monotonic function, can be contrasted with the Pearson correlation, giving a perfect value only when X and Y are related by a linear function. The other reason being, exact sampling distributions can be obtained without requiring knowledge of the joint probability distribution of X and Y (Sheskin, 2003). The Spearman correlation coefficient is based on the assumption that both the predictor and response variables have numeric values, this assumption, however, the Spearman correlation coefficient can be used to analyze variables that are markedly skewed. The Spearman correlation...