This document discusses orthogonal trajectories and provides examples of finding orthogonal trajectories for different families of curves. It begins by defining orthogonal trajectories as curves that intersect each other at right angles. It then provides a method for finding the differential equation that describes the orthogonal trajectories for a given family of curves. Several examples are worked out, such as finding the orthogonal trajectories of the family of parabolas with equation y = x^2. Applications to equipotential lines and electric fields and electromagnetic waves are also mentioned.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
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Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
3. Definition
• An orthogonal trajectory of a family of curves is a curve
that intersects each curve of the family orthogonally—
that is, at right angles.
4. Suppose that we have a family of curves given by
and another family of curves
𝑔 𝑥, 𝑦 = 0
such that at any intersection of the curves of the first
family with a curve of the second family, the tangents of
the curves are perpendicular.
Orthogonal trajectories, therefore, are two families of
curves that always intersect perpendicularly.
𝑓 𝑥, 𝑦 = 0
5. Let us consider the example of the two families
𝑦 = 𝑚𝑥 𝑎𝑛𝑑 𝑦2 + 𝑥2 = 𝑐2
If we draw the two families together on the same graph
we get
6. METHOD
( , )
dy
f x y
dx
1
( , )
dy
dx f x y
Find the differential equation for the given family of
curves, by differentiating
Find the differential equation of the Orthogonal
Trajectory
Separating variables and integrating the above differential
equation we get the algebraic equation of the family of
orthogonal trajectories.
Given family of curve 𝑓 𝑥, 𝑦 = 0
STEP 1
STEP 2
STEP 3
𝑑𝑥
𝑑𝑦
= −𝑓( 𝑥, 𝑦)or
7. Orthogonal Curves
2
Consider the family of parabola . Find the family of curves
which intersect the above family of parabola perpendicularly.
y x C
Example
Solution
Two curve intersect perpendicularly if the product of the slopes of
the tangents at the intersection point is -1.
The differential equation for the orthogonal family of curves.
𝑑𝑦
𝑑𝑥
= −
1
2𝑥
By differentiation we get
𝑑𝑦
𝑑𝑥
= 2𝑥 . Hence the family of
parabola in question satisfies the differential equation
𝑑𝑦
𝑑𝑥
= 2𝑥.
8.
1 1 1
2 2
dy
dy dx
dx x x
The figure on the right shows these
two orthogonal families of curves.
1
2
dy
dx x
1
ln
2
y x C
9. Example Find the orthogonal trajectories of all parabolas with vertices at
the origin and foci on the x-axis
𝒚 𝟐
= 𝟒𝒂𝒙
𝒚 𝟐
𝒙
= 𝟒𝒂
STEP 1: Differentiate Eq 1
Eq 1
𝒙
𝒅
𝒅𝒙
𝒚 𝟐
− 𝒚 𝟐 𝒅
𝒅𝒙
(𝒙)
𝒙 𝟐
= 𝟎
𝟐𝒙𝒚
𝒅𝒚
𝒅𝒙
− 𝒚 𝟐
= 𝟎
𝒅𝒚
𝒅𝒙
=
𝒚
𝟐𝒙
STEP 2: Differential equation of the orthogonal trajectories
𝒅𝒚
𝒅𝒙
= −
𝟐𝒙
𝒚
13. • Electromagnetic waves consist of both electric and magnetic
field waves. These waves oscillate in perpendicular planes with
respect to each other.
14. • The light rays of sun that passes through the
orbits of the planets.