Systems of ODEs
First-order linear equations with constant coefficients
[pic]
[pic]

Let [pic]

[pic]

Taking Laplace transforms of (1) and (2)

[pic]

[pic]

From (3) and (4)

[pic]

[pic]

We solve this system algebraically for [pic]and [pic] and obtain [pic] by taking inverse transforms.

Example [pic]
[pic]

[pic]

We have

[pic]

[pic]
[pic]

From (5) and (6)

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Since[pic]

[pic]

[pic]

From (II)

[pic]

[pic]

[pic]

[pic]

[pic]

Since[pic]

[pic]

Solution:
[pic]
[pic]

Example

[pic]
Advanced Engineering Mathematics (Kreyszig)

Tank T1 initially contains 100 gal of pure water.
Tank T2 initially contains 100 gal of water in which 150lb of salt are dissolved

The inflow into T1 from outside is 6 gal/min with 6 lb of salt. Obtain the salt content [pic] and [pic] in T1 and T2 respectively. Assume that the mixtures are kept uniform.

The above mechanical system consists of two bodies of mass 1 on 3 springs of the same spring constant k. The masses of the springs are negligible. When the system is in static equilibrium, the masses m1 and m2 are at positions A and B respectively, and the springs all have extension zero as the masses of m1 and m2 are small compared with the spring constant. Each spring is neither extended nor compressed.

With the masses in the positions C and D as shown (above right),

Extension of top spring = [pic]

Extension of middle spring = DC – AB
= (PD – PC) – AB
= [(l1 + l2 + y2) – (l1 + y1)] – l2
= y2 – y1

Consider mass x acc. = Force(mass = 1)

For a stretched spring F = - k x (where k is the spring constant and x is the extension)

...Bioresource Technology 100 (2009) 1192–1196
Contents lists available at ScienceDirect
Bioresource Technology
journal homepage: www.elsevier.com/locate/biortech
Optimization of culture conditions for hydrogen production by Ethanoligenens
harbinense B49 using response surface methodology
Wan-Qian Guo a, Nan-Qi Ren a,*, Xiang-Jing Wang b, Wen-Sheng Xiang b, Jie Ding a, Yang You a,
Bing-Feng Liu a
a
b
State Key Laboratory of Urban Water Resource and Environment, Harbin Institute of Technology, Harbin 150090, PR China
Research Center of Life Science and Biotechnology, Northeast Agricultural University, Harbin 150030, PR China
a r t i c l e
i n f o
Article history:
Received 16 May 2008
Received in revised form 28 July 2008
Accepted 28 July 2008
Available online 14 September 2008
Keywords:
Hydrogen production
Ethanol fermentation
Response surface methodology
Ethanoligenens harbinense B49
Culture conditions
a b s t r a c t
The design of an optimum and cost-efﬁcient medium for high-level production of hydrogen by Ethanoligenens harbinense B49 was attempted by using response surface methodology (RSM). Based on the Plackett–Burman design, Fe2+ and Mg2+ were selected as the most critical nutrient salts. Subsequently, the
optimum combination of the selected factors and the sole carbon source glucose were investigated by
the Box–Behnken design. Results showed that the maximum hydrogen yield of 2.21 mol/mol glucose...

...Mitchell v. Lovington Good Samaritan Center, Inc., 555 P.2d 696 (1976).
Facts: The appellee was terminated from the Lovington Good Samaritan Center, Inc. on June 4, 1974. On June 12, 1974 Mrs. Mitchell applied for unemployment compensation benefits. She was initially disqualified from seven weeks of benefits by a deputy of the Unemployment Security Commission. Mrs. Mitchell then filed an appeal, and the Appeal Tribunal reversed the deputy’s decision. Mrs. Mitchell’s benefits were reinstated on August 28, 1974. On September 13, 1974 the Center appealed the decision made by the Appeal tribunal to the whole Commission. The Commission overruled the Appeal Tribunal and reinstated the seven week disqualification period. Mrs. Mitchell then applied for and was granted certiorari from the decision of the Commission to the District Court of Bernalillo County. The District Court reversed the Commission’s decision and ordered the benefits to be reinstated. From the judgment of the District Court, the Center appeals.
Issue: The issue is whether Mrs. Mitchell’s actions constituted misconduct under § 59-9-5(b), N.M.S.A. 1953.
Rule: The term ‘misconduct’ is not defined in the Unemployment Compensation Law. The Wisconsin Supreme Court found that in a prior case no statutory definition of misconduct existed, and they formulated a definition for such. The Supreme Court of New Mexico adopts that definition, and hold that Mrs. Mitchell’s acts constituted misconduct.
Applying: Mrs....

...Northcentral University
Assignment Cover Sheet
Felix Rogombe
SKS5000-8 Comprehensive Strategic Knowledge Studies
5 Paper–Business Economics
Dr. Jaime Klein
Date: February 12, 2012
To: ABC Complete Kitchens, Inc.
From: Felix Rogombe
Attn.: ABC Complete Kitchens, Inc. President and CEO
RE: Economic factors that may influence the plant’s operations and legal considerations.
On a macroeconomic level, according to a United Nations’ study, “The recovery of the global economy continues, with strong output growth in developing countries and weaker economic performance in developed countries.” (UN, 2011)Even so higher energy and food cost are contributing to rising inflation rates. These indicators are mitigated by a stronger world trade of goods and services in 2010 compared to the year of 2009, it should be noted that Asian countries account for most of this growth. According to the above cited UN report, some of the risks witch may hamper the growth that is currently taking place may be the problem regarding the sustainability of public finance of developed countries, including the collapse of the dollar due to continued volatility of commodity prices. This doesn’t take in consideration the unpredictable area of policy making which outcomes cannot be accurately predicted.
Because the financial situation is more favorable internationally ABC Complete Kitchens, Inc. should consider capitalizing on the international market while at the same time mitigating its...

...is cooled
by radiation and convection to its surroundings.
I decided to use 4th order Runge Kutta method to
solve the problem.
Objective
To solve the first order nonlinear ordinary
differential equations using numerical
method.
To understand the 4th order Runge Kutta
method and its applications.
Problem statement
Steel ball bearing radius 0.02m, ρ =
dT
7800kg/m3
A T 4 Ta4 mC
dt
The radiation equation is
The convection equation is
Assumed T0= 1200K and Ambient
temperature,
Assumed that all heat transfer in
radiation and convection only.
Mathematical model
Combining both convection and radiation
equation to form a new rate of heat lost
equation:
After subsitution of the constants, the
equation reduced to:
Preliminary Solution
Rewrite the equations:
=(((-2.20673*(10^-13))*(T^4))-((1.60256*(10^-2))*(T))
+(4.8095))
f(t,T)=(((-2.20673*(10^-13))*(T^4))-((1.60256*(10^-2))*(T))
+(4.8095))
Set
Runge Kutta 4th Order Solution
1
Ti 1 Ti k1 2k 2 2k3 k 4 h
6
k1 f (ti , Ti )
1
1
k 2 f (ti h, Ti k1h)
2
2
1
1
k3 f (ti h, Ti k 2 h)
2
2
k 4 f (ti h, Ti k3 h)
MATLAB IMPLEMENTATION
Result
Discussion and conclusion
The steel ball bearing will reach temperature
1000K at 15.3 seconds
The time taken for the ball bearing to cool from
1200K to 1000K is obtained by using the 4 th
order Runge Kutta method to analyze...

...Numerical Methods Questions
1 f(x) = x3 – 2x – 5
a) Show that there is a root β of f(x) = 0 in the interval [2,3]. The root β is to be estimated using the iterative formula
,2 5 2 0 2 1 1 = =++ x x x n n
b) Calculate the values of x1, x2, x3, and x4, giving your answers to 4 sig fig.
c) Prove that, to 5 significant figures, β is 2.0946
2 Use the iterative formula
n
n n cox x x − =+ 1 1 With x1 = 0.5 to find the limit of the sequence x1, x2, x3,……. Correct to 2 decimal places.
3 Starting with x0 = 1, use the iterative formula
( ) 5lnln1 =++ xx n
To find, to 2 decimal places, the value of x1, x2, x3, and x4.
4 The equation 2x = x3 has two roots. Show that the intervals are [1,2] and [9,10].
5 f( x ) = x 3 – 2 –
x 1
, x ≠ 0.
(a) Show that the equation f( x ) = 0 has a root between 1 and 2.
An approximation for this root is found using the iteration formula
x n + 1 =
3 1
nx 1 2
+ , with 0 x = 1.5.
(b) By calculating the values of x 1, x 2, x 3 and x 4, find an approximation to this root, giving your answer to 3 decimal places.
(c) By considering the change of sign of f( x ) in a suitable interval, verify that your answer to part (b) is correct to 3 decimal places.
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical...

............... 48 3.2.2 Evolution of Online Shopping Behavior.................................................... 50 3.2.3 Usage Experience with Decision Aids and Online Shopping Behavior Evolution ............................................................................................................. 51 3.2.4 Potential Hidden States of Purchase Behavior........................................... 53 3.3 Model Formulation ........................................................................................... 54 3.3.1 Type-II Tobit Model of Category Purchase Incidence and Quantity ........ 54 3.3.2 Hidden States and Transition Probabilities ................................................ 56 3.3.3 Prior Distributions and Estimation Method ............................................... 58 3.4 Empirical Analyses ........................................................................................... 59 3.4.1 Data Description ........................................................................................ 59 iii
3.4.2 Operationalization of Key Variables ......................................................... 60 3.4.3 Time-Varying Patterns of Usage Experience with Decision Aids............. 61 3.4.4 Model Estimation Results .......................................................................... 62 3.4.5 Evolution of Purchase Behavior in the Online Store ................................. 64 3.5 Discussion...

...references to relevant items in the category frame, per chosen unit of content.
- Express the results as an overall distribution of the chosen content sample in terms of the frequency of occurrence of the sought-for referents.
This procedure is based on two main assumptions: first, that the link between the external object of reference and the reference to it in the text will be reasonably clear and unambiguous, and second, that the frequency of occurrence of chosen references will validly express the predominant “meaning” of the text in an objective way. The method produces a statistical summary of a much larger media reality, it has been used especially for comparing media content with a known frequency distribution in “social reality”.
LIMITs 362
ARTHUR ASA BERGER: Media Research Techniques !!!!???? Missing!!!!
“ Media Analysis Techniques
97 C.A. is a standard sociological technique that uses statistical methods to make inferences about what is found in the media (:97).
-Historical frame of ref to make comparisons between earlier samples, is there more or less X? if so, what does it mean?
Topic: the representation of young men in Irish advertising.
The topic only addresses age and gender: young men, it does not specify any research questions or context, which leaves the analyst not being certain about which are the variables or measurable elements of interest (Hornig:87). The field is advertising, regarding representations, no context...

...Two very important competencies for health professionals in problem solving & decision making.there are 5 steps on problem solving and decision making.
Step 1:
Now we are going to identifying the problem. It's not always simple as it sounds. Identifying the real nature of problems requires a willingness to observe, pay attention,and comfort difficult issues. Denying problems does not make them go away. Because , if you don't address them, the problem tend to get worse.having take any action to resolve them. It turns out that the real problem is actually what caused the poor evaluation, not the evaluation it self.this is why accepting responsibility for a problem makes it possible to start going something about it.
Step2:
Accurate and reliable information , for you to solve a problem. Acting on assumption,believe not based on facts or knowlege.there are many method for gathering information, review what is already know
Collect the objective data signs
Collect subjective data,also know as symptoms. Conduct research:what are the facts, are they reliable, are they scientifically based.as for help:who has useful knowlege
Step3:
Create a list if alternative is the third step in problem solving.generate ideas for solutions and actions, based on the information collected. All possibilities should be considered before one is selected.It is essential to think though each one and consider...

347 Words |
2 Pages

Share this Document

{"hostname":"studymode.com","essaysImgCdnUrl":"\/\/images-study.netdna-ssl.com\/pi\/","useDefaultThumbs":true,"defaultThumbImgs":["\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_1.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_2.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_3.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_4.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_5.png"],"thumb_default_size":"160x220","thumb_ac_size":"80x110","isPayOrJoin":false,"essayUpload":false,"site_id":1,"autoComplete":false,"isPremiumCountry":false,"userCountryCode":"US","logPixelPath":"\/\/www.smhpix.com\/pixel.gif","tracking_url":"\/\/www.smhpix.com\/pixel.gif","cookies":{"unlimitedBanner":"off"},"essay":{"essayId":37459921,"categoryName":"Periodicals","categoryParentId":"17","currentPage":1,"format":"text","pageMeta":{"text":{"startPage":1,"endPage":6,"pageRange":"1-6","totalPages":6}},"access":"premium","title":"Solving Pde Using Method of Characteristics","additionalIds":[19,2,3,13],"additional":["Natural Sciences","Awards \u0026 Events","Business \u0026 Economy","Health \u0026 Medicine"],"loadedPages":{"html":[],"text":[1,2,3,4,5,6]}},"user":null,"canonicalUrl":"http:\/\/www.studymode.com\/course-notes\/Solving-Pde-Using-Method-Of-Characteristics-1560202.html","pagesPerLoad":50,"userType":"member_guest","ct":10,"ndocs":"1,500,000","pdocs":"6,000","cc":"10_PERCENT_1MO_AND_6MO","signUpUrl":"https:\/\/www.studymode.com\/signup\/","joinUrl":"https:\/\/www.studymode.com\/join","payPlanUrl":"\/checkout\/pay","upgradeUrl":"\/checkout\/upgrade","freeTrialUrl":"https:\/\/www.studymode.com\/signup\/?redirectUrl=https%3A%2F%2Fwww.studymode.com%2Fcheckout%2Fpay%2Ffree-trial\u0026bypassPaymentPage=1","showModal":"get-access","showModalUrl":"https:\/\/www.studymode.com\/signup\/?redirectUrl=https%3A%2F%2Fwww.studymode.com%2Fjoin","joinFreeUrl":"\/essays\/?newuser=1","siteId":1,"facebook":{"clientId":"306058689489023","version":"v2.8","language":"en_US"}}