1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
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Find the explicit solution of the linear DE dyxdx=-6x^3-6x^2 y+1 u.pdfsales89
Find the explicit solution of the linear DE dy/xdx=-6/x^3-6x^2 y+1 using the appropriate
integrating factor and with the initial value of y(10)=-15. (b)Find the largest interval of definition
I for x. (c)Which of the terms in the solution are transient(show limits)?
Solution
A differential equation is a mathematicalequation for an unknown function of one
or several variables that relates the values of the function itself and its derivatives of various
orders. Differential equations play a prominent role in engineering, physics, economics, and
other disciplines. Differential equations arise in many areas of science and technology,
specifically whenever a deterministic relation involving some continuously varying quantities
(modeled by functions) and their rates of change in space and/or time (expressed as derivatives)
is known or postulated. This is illustrated in classical mechanics, where the motion of a body is
described by its position and velocity as the time varies. Newton\'s laws allow one to relate the
position, velocity, acceleration and various forces acting on the body and state this relation as a
differential equation for the unknown position of the body as a function of time. In some cases,
this differential equation (called an equation of motion) may be solved explicitly. An example of
modelling a real world problem using differential equations is determination of the velocity of a
ball falling through the air, considering only gravity and air resistance. The ball\'s acceleration
towards the ground is the acceleration due to gravity minus the deceleration due to air resistance.
Gravity is constant but air resistance may be modelled as proportional to the ball\'s velocity. This
means the ball\'s acceleration, which is the derivative of its velocity, depends on the velocity.
Finding the velocity as a function of time involves solving a differential equation. Differential
equations are mathematically studied from several different perspectives, mostly concerned with
their solutions—the set of functions that satisfy the equation. Only the simplest differential
equations admit solutions given by explicit formulas; however, some properties of solutions of a
given differential equation may be determined without finding their exact form. If a self-
contained formula for the solution is not available, the solution may be numerically
approximated using computers. The theory of dynamical systems puts emphasis on qualitative
analysis of systems described by differential equations, while many numerical methods have
been developed to determine solutions with a given degree of accuracy. The term homogeneous
differential equation has several distinct meanings. One meaning is that a first-order ordinary
differential equation is homogeneous (of degree 0) if it has the form \\frac{dy}{dx} = F(x,y)
where F(x,y) is a homogeneous function of degree zero; that is to say, such that F(tx,ty) = F(x,y).
In a related, but distinct, usage, the term linear .
On the Numerical Fixed Point Iterative Methods of Solution for the Boundary V...BRNSS Publication Hub
In this research work, we have studied the finite difference method and used it to solve elliptic partial differential equation (PDE). The effect of the mesh size on typical elliptic PDE has been investigated. The effect of tolerance on the numerical methods used, speed of convergence, and number of iterations was also examined. Three different elliptic PDE’s; the Laplace’s equation, Poisons equation with the linear inhomogeneous term, and Poisons equations with non-linear inhomogeneous term were used in the study. Computer program was written and implemented in MATLAB to carry out lengthy calculations. It was found that the application of the finite difference methods to an elliptic PDE transforms the PDE to a system of algebraic equations whose coefficient matrix has a block tri-diagonal form. The analysis carried out shows that the accuracy of solutions increases as the mesh is decreased and that the solutions are affected by round off errors. The accuracy of solutions increases as the number of the iterations increases, also the more efficient iterative method to use is the SOR method due to its high degree of accuracy and speed of convergence
It shows the basic facts of catalyst along with its importance in industry along with its long last milestone,its characteristics & application in industry its reaction process and preparation of a solid catalyst.
Production of biodiesel from jatropha plantNofal Umair
Production of Bio-diesel from jatropha plant ....
By the increase in demand of fuel the resources are not as many to full control the demand of the world and the known reservoir wont last forever there fore an alternate energy source is required to fulfill the world fuel demand.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
3. Differential Equations
An equation which involves unknown function of one or several variables that
relates the values of the function itself and its derivatives of various orders.
ordinary differential equation (ode) : not involve partial derivatives
partial differential equation (pde) : involves partial derivatives
order of the differential equation is the order of the highest derivatives
Examples:
second order ordinary differential
equation
first order partial differential equation
2
2
3 sin
d y dy
x y
dxdx
y y x t
x
t x x t
4. Terminologies In Differential
Equation
• Existence: Does a differential equation have a
solution?
• Uniqueness: Does a differential equation have more
than one solution? If yes, how can we find a solution
which satisfies particular conditions?
• A problem in which we are looking for the unknown
function of a differential equation where the values of
the unknown function and its derivatives at some point
are known is called an initial value problem (in short
IVP).
• If no initial conditions are given, we call the description
of all solutions to the differential equation the general
solution.
6. Linear Differential Equation
A differential equation is linear, if
1. dependent variable and its derivatives are of degree one,
2. coefficients of a term does not depend upon dependent
variable.
Example:
36
4
3
3
y
dx
dy
dx
yd
is non - linear because in 2nd term is not of degree one.
.0932
2
y
dx
dy
dx
ydExample:
is linear.
1.
2.
8. First Order Linear Equations
• A linear first order equation is an equation
that can be expressed in the form
Where P and Q are functions of x
9. History
YEAR PROBLEM DESCRIPTION MATHAMATICIAN
1690 Problem of the
Isochrones
Finding a curve
along which a body
will fall with uniform
vertical velocity
James Bernoulli
1728 Problem of
Reducing 2nd Order
Equations to 1st
Order
Finding an
integrating factor
Leonhard Euler
1743 Problem of
determining
integrating factor for
the general linear
equation
Concept of the ad-
Joint of a differential
equation
Joseph Lagrange
1762 Problem of Linear
Equation with
Constant
Coefficients
Conditions under
which the order of a
linear differential
equation could be
lowered
Jean d’Alembert
10. Methods Solving LDE
1. Separable variable
M(x)dx + N(y)dy = 0
2. Homogenous
M(x,y)dx+N(x,y)dy=0, where M & N are nth degree
3. Exact
M(x,y)dx + N(x,y)dy=0, where M/ðy=0, where ðM/ðy = ðN/ðx
12. 1st Order DE - Separable Equations
The differential equation M(x,y)dx + N(x,y)dy = 0 is separable if the equation can
be written in the form:
02211 dyygxfdxygxf
Solution :
1. Multiply the equation by integrating factor:
ygxf 12
1
2. The variable are separated :
0
1
2
2
1
dy
yg
yg
dx
xf
xf
3. Integrating to find the solution:
Cdy
yg
yg
dx
xf
xf
1
2
2
1
13. 1st Order DE - Homogeneous Equations
Homogeneous Function
f (x,y) is called homogenous of degree n if :
y,xfy,xf n
Examples:
yxxy,xf 34 homogeneous of degree 4
yxfyxx
yxxyxf
,
,
4344
34
yxxyxf cossin, 2 non-homogeneous
yxf
yxx
yxxyxf
n
,
cossin
cossin,
22
2
14. 1st Order DE - Homogeneous Equations
The differential equation M(x,y)dx + N(x,y)dy = 0 is homogeneous if M(x,y) and
N(x,y) are homogeneous and of the same degree
Solution :
1. Use the transformation to : dvxdxvdyvxy
2. The equation become separable equation:
0,, dvvxQdxvxP
3. Use solution method for separable equation
Cdv
vg
vg
dx
xf
xf
1
2
2
1
4. After integrating, v is replaced by y/x
15. 1st Order DE – Exact Equation
The differential equation M(x,y)dx + N(x,y)dy = 0 is an exact equation if :
Solution :
The solutions are given by the implicit equation
x
N
y
M
CyxF ,
1. Integrate either M(x,y) with respect to x or N(x,y) to y.
Assume integrating M(x,y), then :
where : F/ x = M(x,y) and F/ y = N(x,y)
ydxyxMyxF ,,
2. Now : yxNydxyxM
yy
F
,',
or : dxyxM
y
yxNy ,,'
16. 1st Order DE – Exact Equation
3. Integrate ’(y) to get (y) and write down the result F(x,y) = C
Examples:
1. Solve :
01332 3
dyyxdxyx
Answer:
17. Newton's Law of Cooling
• It is a model that describes, mathematically, the change in temperature of
an object in a given environment. The law states that the rate of change (in
time) of the temperature is proportional to the difference between the
temperature T of the object and the temperature Te of the environment
surrounding the object.
d T / d t = - k (T - Te)
Let x = T - Te
so that dx / dt = dT / dt
d x / d t = - k x
The solution to the above differential equation is given by
x = A e - k t
substitute x by T – Te
T - Te = A e - k t
Assume that at t = 0 the temperature T = To
18. T0 - Te = A e o
which gives A = To-Te
The final expression for T(t) is given by T(t) = Te + (To- Te) e - k t
This last expression shows how the temperature T of the object changes with time.
19. Growth And Decay
• The initial value problem
where N(t) denotes population at time t and k is a constant of proportionality,
serves as a model for population growth and decay of insects, animals and
human population at certain places and duration.
Integrating both sides we get
ln N(t)=kt+ln C
or
or N(t)=Cekt
C can be determined if N(t) is given at certain time.
)(
)(
tkN
dt
tdN
kdt
tN
tdN
)(
)(
20. Carbon dating
Let M(t) be the amount of a product that decreases withtime t and the rate of
decrease is proportional to the amount M as follows
d M / d t = - k M
where d M / d t is the first derivative of M, k > 0 and t is the time.
Solve the above first order differential equation to obtain
M(t) = Ae-kt
where A is non zero constant. It we assume that M = Mo at t = 0, then
M= Ae0
which gives A = Mo
The solution may be written as follows
M(t) = Mo e-kt
21. Economics and Finance
• The problems regarding supply, demand and compounding interest can be
calculated by this equation
is a separable differential equation of first-order. We can write it as
dP=k(D-S) dt.
Integrating both sides, we get
P(t)=k(D-S)t+A
where A is a constant of integration.
Similarly
S(t)=S(0) ert ,Where S(0) is the initial money in the account
)( SDk
dt
dP