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Differential equations relate functions and their derivatives. Ordinary differential equations involve one independent variable and one or more dependent variables and their derivatives. Partial differential equations involve more than one independent variable. The order of a differential equation is defined as the highest derivative present. Differential equations have many applications across sciences like physics, biology, chemistry, and mathematics to model real-world phenomena involving rates of change.

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Differential Equations

Differential Equations

Differential Equations: All about it's order, degree, application and more | ...

Differential Equations: All about it's order, degree, application and more | ...

Differential equations

Differential equations

Report

Share

Differential Equations

Differential equations describe relationships between variables and their derivatives. There are various types of differential equations classified by order and degree. The order refers to the highest derivative present, while the degree refers to the power of the highest derivative. Differential equations have many applications, including modeling population growth, economics, physics, engineering, and more.

Differential Equations: All about it's order, degree, application and more | ...

Fear of differential equations questions? With the usage of explicit formulas given in this article, you can easily solve this type of equation.

Differential equations

This document defines and provides examples of different types of differential equations. It discusses ordinary differential equations that depend on one independent variable and partial differential equations that depend on two or more variables. It also defines linear and nonlinear differential equations, homogeneous and non-homogeneous equations, and describes solutions including general, particular, and singular solutions. Applications of differential equations discussed include physics, geometry, exponential growth, exponential decay, and Newton's law of cooling.

Find the explicit solution of the linear DE dyxdx=-6x^3-6x^2 y+1 u.pdf

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 solutionsâ€”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 .

Differential equations of first order

1) A differential equation contains an independent variable (x), a dependent variable (y), and the derivative of the dependent variable with respect to the independent variable (dy/dx).
2) The order of a differential equation refers to the highest order derivative present. For example, an equation containing dy/dx would be first order, while one containing d2y/dx2 would be second order.
3) The degree of a differential equation refers to the highest power of the highest order derivative. For example, an equation containing (d2y/dx) would be degree 1, while one containing (d2y/dx)2 would be degree 2.
4) There are several methods for solving first

Basics de

This document provides an overview of differential equations. It defines a differential equation as an equation containing one or more terms and the derivatives of one variable with respect to another. Differential equations can be first order, containing only the first derivative, or second order, containing the second derivative. There are several types of differential equations classified by whether they are ordinary or partial, linear or nonlinear, homogeneous or nonhomogeneous. The document also gives examples of applying differential equations to describe exponential growth/decay, return on investment over time, modeling disease spread, electrical movement, and engineering applications like heat conduction, wave motion, and structural analysis.

11Functions-in-several-variables-and-double-integrals.pdf

This document discusses functions of multiple variables and their properties. It introduces functions of two variables and their graphs, which can be interpreted as surfaces in space. It then discusses partial derivatives, which are derivatives of functions of multiple variables where one variable is held constant. Higher order partial derivatives and double integrals are also introduced. An example problem finds the volume of a solid bounded by a paraboloid and planes using iterated integrals.

Numerical_PDE_Paper

This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.

First Order Ordinary Differential Equations FAHAD SHAHID.pptx

numerical analysis course
presentation on topic first order ordinary differential equation
for student to help in their study

microproject@math (1).pdf

This document discusses differential equations. It defines ordinary and partial differential equations, and explains that ordinary differential equations involve functions of a single variable while partial differential equations involve functions of multiple variables. It also defines linear and non-linear differential equations, and discusses methods for solving different types of differential equations including separable, exact, and linear equations with constant or variable coefficients. Examples are provided for population growth models, price dynamics, and economic models involving differential equations.

Applications of partial differentiation

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

M1 unit i-jntuworld

The document contains information about the contents, textbooks, and references for the Mathematics-I course. It outlines 15 lectures covering various topics in ordinary differential equations of first order and first degree, including exact differential equations, integrating factors, linear and Bernoulli's equations, orthogonal trajectories, Newton's law of cooling, and laws of natural growth and decay. Examples are provided to illustrate key concepts discussed in each lecture.

download-1679883048969u.pptx

Differential equations relate a function with its derivatives. They model systems where quantities change, with derivatives representing rates of change. There are two main types: ordinary differential equations containing ordinary derivatives of a single variable, and partial differential equations with partial derivatives of multiple variables. Differential equations are classified by order, degree, linearity and type. Solutions are determined using general solutions along with initial or boundary conditions.

Applications of differential equation

1. The document discusses differential equations and their applications. It defines differential equations and describes their use in fields like physics, engineering, biology and economics to model complex systems.
2. Examples of first order differential equations are given to model exponential growth, exponential decay, and an RL circuit. Higher order differential equations are used to model falling objects and Newton's law of cooling.
3. The key applications covered are population growth, radioactive decay, free falling objects, heat transfer, and electric circuits. Solving the differential equations gives mathematical models relating variables like position, temperature, and current over time.

Polynomials

The document discusses polynomials. It defines polynomials as expressions constructed from variables and constants using addition, subtraction, multiplication, and non-negative integer exponents. It provides examples of polynomials and non-polynomial expressions. It also discusses the degrees of terms and polynomials, and how polynomials can be added or multiplied by distributing terms. The document also covers monomials, binomials, trinomials, and the factor theorem.

ORDINARY DIFFERENTIAL EQUATION

1) The document presents information on ordinary differential equations including definitions, types, order, degree, and solution methods.
2) Differential equations can be written in derivative, differential, and differential operator forms. Common solution methods covered are variable separable, homogeneous, linear, and exact differential equations.
3) Applications of differential equations include physics, astronomy, meteorology, chemistry, biology, ecology, and economics for modeling various real-world systems.

Ft3 new

1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.

Calculus

This document discusses the difference between ordinary derivatives and partial derivatives. Ordinary derivatives are taken with respect to one independent variable, while partial derivatives treat all other variables as constant except for the variable being differentiated. The document provides examples of calculating ordinary and partial derivatives for functions of two variables. It also notes that partial derivatives are used in the chain rule to compute ordinary derivatives for functions with multiple variables.

Maths differential equation ppt

The document defines and provides examples of differential equations. The key points are:
- A differential equation is an equation involving one or more independent variables, one dependent variable, and the derivatives of the dependent variable.
- Differential equations can be ordinary (contain one independent variable) or partial (contain two or more independent variables).
- The order of a differential equation is the order of the highest derivative. The degree refers to the highest power of the highest derivative when expressed as a polynomial.
- The general solution of a differential equation contains as many arbitrary constants as the order of the equation. A particular solution assigns specific values to the constants.

Limits and derivatives

The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.

Differential Equations

Differential Equations

Differential Equations: All about it's order, degree, application and more | ...

Differential Equations: All about it's order, degree, application and more | ...

Differential equations

Differential equations

Find the explicit solution of the linear DE dyxdx=-6x^3-6x^2 y+1 u.pdf

Find the explicit solution of the linear DE dyxdx=-6x^3-6x^2 y+1 u.pdf

Differential equations of first order

Differential equations of first order

Basics de

Basics de

11Functions-in-several-variables-and-double-integrals.pdf

11Functions-in-several-variables-and-double-integrals.pdf

Numerical_PDE_Paper

Numerical_PDE_Paper

First Order Ordinary Differential Equations FAHAD SHAHID.pptx

First Order Ordinary Differential Equations FAHAD SHAHID.pptx

microproject@math (1).pdf

microproject@math (1).pdf

Applications of partial differentiation

Applications of partial differentiation

M1 unit i-jntuworld

M1 unit i-jntuworld

download-1679883048969u.pptx

download-1679883048969u.pptx

Applications of differential equation

Applications of differential equation

Polynomials

Polynomials

ORDINARY DIFFERENTIAL EQUATION

ORDINARY DIFFERENTIAL EQUATION

Ft3 new

Ft3 new

Calculus

Calculus

Maths differential equation ppt

Maths differential equation ppt

Limits and derivatives

Limits and derivatives

Potential of Marine renewable and Non renewable energy.pptx

To study about Potential of Marine renewable energy and non renewable energy

Simulations of pulsed overpressure jets: formation of bellows and ripples in ...

Jets from active nuclei may supply the heating which moderates cooling and accretion from the circum-galactic medium. While
steady overpressured jets can drive a circulatory flow, lateral energy transfer rarely exceeds 3 per cent of jet power, after the initial
bow shock has advanced. Here, we explore if pulses in high-pressure jets are capable of sufficient lateral energy transfer into
the surrounding environment. We answer this by performing a systematic survey of numerical simulations in an axisymmetric
hydrodynamic mode. Velocity pulses along low Mach jets are studied at various overpressures. We consider combinations of
jet velocity pulse amplitude and frequency. We find three flow types corresponding to slow, intermediate, and fast pulsations.
Rapid pulsations in light jets generate a series of travelling shocks in the jet. They also create ripples which propagate into the
ambient medium while a slow convection flow brings in ambient gas which is expelled along the jet direction. Long period pulses
produce slowly evolving patterns which have little external effect, while screeching persists as in non-pulsed jets. In addition,
rapid pulses in jets denser than the ambient medium generate a novel breathing cavity analogous to a lung. Intermediate period
pulses generate a series of bows via a bellows action which transfer energy into the ambient gas, reaching power efficiencies of
over 30 per cent when the jet overpressure issufficiently large. This may adequately inhibit galaxy gas accretion. In addition,such
pulses enhance the axial out-flow of jet material, potentially polluting the circum-galactic gas with metal-enriched interstellar
gas.

Lake classification and Morphometry.pptx

To study about classification of lakes and Morphometry

Surface properties of the seas of Titan as revealed by Cassini mission bistat...

Saturn’s moon Titan was explored by the Cassini spacecraft from 2004 to 2017.
While Cassini revealed a lot about this Earth-like world, its radar observations
could only provide limited information about Titan’s liquid hydrocarbons seas
Kraken, Ligeia and Punga Mare. Here, we show the results of the analysis of the
Cassini mission bistatic radar experiments data of Titan’s polar seas. The dualpolarized nature of bistatic radar observations allow independent estimates of
effective relative dielectric constant and small-scale roughness of sea surface,
which were not possible via monostatic radar data. We find statistically significant variations in effective dielectric constant (i.e., liquid composition),
consistent with a latitudinal dependence in the methane-ethane mixing-ratio.
The results on estuaries suggest lower values than the open seas, compatible
with methane-rich rivers entering seas with higher ethane content. We estimate small-scale roughness of a few millimeters from the almost purely
coherent scattering from the sea surface, hinting at the presence of capillary
waves. This roughness is concentrated near estuaries and inter-basin straits,
perhaps indicating active tidal currents.

Complementary interstellar detections from the heliotail

The heliosphere is a protective shield around the solar system created by the Sun’s interaction with the local interstellar medium (LISM) through the solar wind, transients, and interplanetary magnetic field. The shape of the heliosphere is directly linked with interactions with the surrounding LISM, in turn affecting the space environment within the heliosphere. Understanding the shape of the heliosphere, the LISM properties, and their interactions is critical for understanding the impacts within the solar system and for understanding other astrospheres. Understanding the shape of the heliosphere requires an understanding of the heliotail, as the shape is highly dependent upon the heliotail and its LISM interactions. The heliotail additionally presents an opportunity for more direct in situ measurement of interstellar particles from within the heliosphere, given the likelihood of magnetic reconnection and turbulent mixing between the LISM and the heliotail. Measurements in the heliotail should be made of pickup ions, energetic neutral atoms, low energy neutrals, and cosmic rays, as well as interstellar ions that may be injected into the heliosphere through processes such as magnetic reconnection, which can create a direct magnetic link from the LISM into the heliosphere. The Interstellar Probe mission is an ideal opportunity for measurement either along a trajectory passing through the heliotail, via the flank, or by use of a pair of spacecraft that explore the heliosphere both tailward and noseward to yield a more complete picture of the shape of the heliosphere and to help us better understand its interactions with the LISM.

Testing the Son of God Hypothesis (Jesus Christ)

Instead of answering the God hypothesis, we investigate the Son of God hypothesis. We developed our own methodology to deal with existential statements instead of universal statements unlike science. We discuss the existence of the supernaturals and found that there are strong evidence for it. Given that supernatural exists, we report on miracles investigated in the past related to the Son of God. A Bayesian methodology is used to calculate the combined degree of belief of the Son of God Hypothesis. We also report the testing of occurrences of words/numbers in the Bible to suggest the likelihood of some special numbers occurring, supporting the Son of God Hypothesis. We also have a table showing the past occurrences of miracles in hundred year periods for about 1000 years. Miracles that we have looked at include Shroud of Turin, Eucharistic Miracles, Marian Apparitions, Incorruptible Corpses, etc.

20240710 ACMJ Diagrams Set 3.docx . Apache, Csharp, Mysql, Javascript stack a...

Diagrams of made early prototypes of ACMJ components, to do with electricity.

AlgaeBrew project - Unlocking the potential of microalgae for the valorisatio...

AlgaeBrew project - Unlocking the potential of microalgae for the valorisatio...Faculty of Applied Chemistry and Materials Science

AlgaeBrew project - Unlocking the potential of microalgae for the valorisation of brewery waste products into omega-3 rich animal feed and fertilisers
Carmen Gabriela Constantin, University of Agronomic Sciences and Veterinary Medicine (USAMV), RomaniaAnalytical methods for blue residues characterization - Oana Crina Bujor

Analytical methods for blue residues characterization - Oana Crina BujorFaculty of Applied Chemistry and Materials Science

Analytical methods for blue residues characterization
Oana Crina Bujor, University of Agronomic Sciences and Veterinary Medicine (USAMV), RomaniaBurn child health Nursing 3rd year presentation..pptx

Easy learning...pptx..for professional growth and development ❣️🎊

Introduction to Space (Our Solar System)

Space is tremendous, apparently endless span that exists past earth and its environment. It is a locale up with endless heavenly bodies,
including stars, planets, moons, space rocks, and comets, all represented by the gravity. Space investigation has extended how we might interpret the universe, uncovering the excellence and intricacy of far off cosmic system, the secret of dark openings, and the potential for life past our planet. An outskirts keeps of motivating interest, logical request, and a feeling of marvel about our spot in the universe. Space is immense, largely unexplored expanse beyond Earth's atmosphere, home to countless celestial bodies likes stars, planets, and asteroids. Human exploration began with the launch of Sputnik in 1957 followed by significant achievements such as the Moon landing in 1969.

SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...

We present the second data release (DR2) of the Far-Infrared Polarimetric Large-Area CMZ Exploration (FIREPLACE) survey. This survey utilized the Stratospheric Observatory for Infrared Astronomy (SOFIA) High-resolution Airborne Wideband Camera plus (HAWC+) instrument at 214 µm
(E-band) to observe dust polarization throughout the Central Molecular Zone (CMZ) of the Milky
Way. DR2 consists of observations that were obtained in 2022 covering the region of the CMZ extending roughly from the Brick to the Sgr C molecular clouds (corresponding to a roughly 1◦ × 0.75◦
region
of the sky). We combine DR2 with the first FIREPLACE data release covering the Sgr B2 region to
obtain full coverage of the CMZ (a 1.5◦ ×0.75◦
region of the sky). After applying total and polarized
intensity significance cuts on the full FIREPLACE data set we obtain ∼65,000 Nyquist-sampled polarization pseudovectors. The distribution of polarization pseudovectors confirms a bimodal distribution
in the CMZ magnetic field orientations, recovering field components that are oriented predominantly
parallel or perpendicular to the Galactic plane. These magnetic field orientations indicate possible
connections between the previously observed parallel and perpendicular distributions. We also inspect
the magnetic fields toward a set of prominent CMZ molecular clouds (the Brick, Three Little Pigs,
50 km s−1
, Circum-nuclear Disk, CO 0.02-0.02, 20 km s−1
, and Sgr C), revealing spatially varying
magnetic fields that generally trace the morphologies of the clouds. We find evidence that compression
from stellar winds and shear from tidal forces are prominent mechanisms influencing the structure of
the magnetic fields observed within the clouds.

End of pipe treatment: Unlocking the potential of RAS waste - Carlos Octavio ...

End of pipe treatment: Unlocking the potential of RAS waste - Carlos Octavio ...Faculty of Applied Chemistry and Materials Science

End of pipe treatment: Unlocking the potential of RAS waste
Carlos Octavio Letelier-Gordo, DTU Aqua, DenmarkDirect instructions, towards hundred fold yield,layering,budding,grafting,pla...

Fertility, plants, layering, growth, health of seeds

17. 20240529_Ingrid Olesen_MariGreen summer school.pdf

Moving beyond agriculture and aquaculture to integrated sustainable food systems as part of a circular bioeconomy

MARIGREEN PROJECT - overview, Oana Cristina Pârvulescu

MARIGREEN PROJECT - overview, Oana Cristina PârvulescuFaculty of Applied Chemistry and Materials Science

MARIGREEN PROJECT - overview
Oana Cristina Pârvulescu, Project coordinator, POLITEHNICA Bucharest, Romania
All-domain Anomaly Resolution Office Supplement to Oak Ridge National Laborat...

In 2022, The All-domain Anomaly Resolution Office (AARO) contracted with Oak Ridge
National Laboratory (ORNL) to conduct materials testing on a magnesium (Mg) alloy specimen.
This specimen has been publicly alleged to be a component recovered from a crashed
extraterrestrial vehicle in 1947, and purportedly exhibits extraordinary properties, such as
functioning as a terahertz waveguide to generate antigravity capabilities. In April 2024, ORNL
produced a summary of findings documenting the laboratory’s methodology to assess this
specimen’s elemental and structural characteristics, available on AARO’s website.
ORNL assessed this specimen to be terrestrial in origin and that it does not meet the theoretical
requirements to function as a terahertz (THz) waveguide. AARO concurs with ORNL’s
assessment and provides this supplementary material to add historical context to account for its
likely origin. The specimen’s characteristics are consistent with Mg alloy research and
development projects and experimental manufacturing methods in the mid-20th century.

MCQ in Electrostatics. for class XII pptx

Physics Multiple choice questions and answers with explanation. (Class XII Physics TN State board)

Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...

Ascertaining the morphology and composition of the icy mantles covering
dust grains in dense, cold regions of the interstellar medium is essential to
developing accurate astrochemical models, determining conditions for
ice formation, constraining chemical interactions in and on icy grains and
understanding how ices withstand space radiation. The widely observed
infrared spectroscopic signature of H2O ice at ~3 μm discriminates crystalline
from amorphous structures in interstellar ices. Weaker bands seen only in
laboratory ice spectra at ~2.7 μm, termed ‘dangling OH’ (dOH), are attributed
to water molecules not fully bound to neighbouring water molecules and
are often considered as tracing the degree of ice compaction. We exploit
the high sensitivity of JWST NIRCam to detect two dOH features at 2.703 and
2.753 μm along multiple lines of sight probing the dense cloud Chamaeleon
I, attributing these signatures to unbound dOH in cold water ice and dOH in
interaction with other molecular species. These detections open a path to
using the dOH features as tracers of the formation, composition, morphology
and evolution of icy grains during the star and planet formation process.

Potential of Marine renewable and Non renewable energy.pptx

Potential of Marine renewable and Non renewable energy.pptx

Simulations of pulsed overpressure jets: formation of bellows and ripples in ...

Simulations of pulsed overpressure jets: formation of bellows and ripples in ...

Lake classification and Morphometry.pptx

Lake classification and Morphometry.pptx

Surface properties of the seas of Titan as revealed by Cassini mission bistat...

Surface properties of the seas of Titan as revealed by Cassini mission bistat...

Complementary interstellar detections from the heliotail

Complementary interstellar detections from the heliotail

Testing the Son of God Hypothesis (Jesus Christ)

Testing the Son of God Hypothesis (Jesus Christ)

20240710 ACMJ Diagrams Set 3.docx . Apache, Csharp, Mysql, Javascript stack a...

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Adjusted NuGOweek 2024 Ghent programme flyer

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Analytical methods for blue residues characterization - Oana Crina Bujor

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Burn child health Nursing 3rd year presentation..pptx

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Introduction to Space (Our Solar System)

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17. 20240529_Ingrid Olesen_MariGreen summer school.pdf

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MARIGREEN PROJECT - overview, Oana Cristina Pârvulescu

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Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...

- 1. Differential Equations In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. dy/dx = f(x) Here “x” is an independent variable and “y” is a dependent variable There are a lot ofdifferential equation formulas to find the solution of the derivatives. Differential equations can be divided into several types namely •Ordinary Differential Equations •Partial Differential Equations •Linear Differential Equations •Non-linear differential equations
- 2. Ordinary Differential Equation An ordinary differential equation involves function and its derivatives. It contains only one independent variable and one or more of its derivative with respect to the variable. The order of ordinary differential equations is defined as the order of the highest derivative that occurs in the equation. The general form of n-th order ODE is given as F(x, y, y’,…., yn ) = 0 he term “Ordinary Differential Equations” also known as ODE Example 1: Find the solution to the ordinary differential equation y’=2x+1 Solution: Given, y’=2x+1 Now integrate on both sides, ∫ y’dx = ∫ (2x+1)dx y = 2x2 /2 + x + C y =2𝑥2 + x + C Where C is an arbitrary constant
- 3. Partial Differential Equation A Partial Differential Equation commonly denoted as PDE is a differential equation containing partial derivatives of the dependent variable (one or more) with more than one independent variable. A PDE for a function u(x1,……xn) is an equation of the form The PDE is said to be linear if f is a linear function of u and its derivatives. The simple PDE is given by; ∂u/∂x (x,y) = 0 The above relation implies that the function u(x,y) is independent of x which is the reduced form of partial differential equation formula stated above. The order of PDE is the order of the highest derivative term of the equation.
- 4. How to Represent Partial Differential Equation? In PDEs, we denote the partial derivatives using subscripts, such as; In some cases, like in Physics when we learn about wave equations or sound equation, partial derivative, ∂ is also represented by ∇(del or nabla).
- 5. Order Of The Differential Equation The order of the differential equation is the order of the highest order derivative present in the equation. Here some examples for different orders of the differential equation are g iven. •. dy/dx = 3x + 2 , The order of the equation is 1 •(d2y/dx2)+ 2 (dy/dx)+y = 0. The order is 2 •(dy/dt)+y = kt. The order is 1 First Order Differential Equation You can see in the first example, it is a first-order differential equation which has degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’ Second-Order Differential Equation The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”
- 6. Higher Order Derivatives the derivative of a function y = f( x) is itself a function y′ = f′( x), you can take the derivative of f′( x), which is generally referred to as the second derivative of f(x) and written f“( x) or f 2( x). This differentiation process can be continued to find the third, fourth, and successive derivatives of f( x), which are called higher order derivatives f( n )( x) = y( n ) to denote the nth derivative of f( x).
- 7. Differential Equations Applications Differential Equation applications have significance in both academic and real life. An equation denotes the relation between two quantity or two functions or two variables or set of variables or between two functions. Differential equation denotes the relationship between a function and its derivatives, with some set of formulas. There are many examples, which signifies the use of these equations. Applications in various science field 1.In physics a. The derivative of displacement of a moving body with respect to time ,is the velocity of the body and the derivative of velocity W.r.t time is acceleration. b. Newton second law of motion state that ,the derivative of the momentum of body is equal to the force applied to the body. 2. In biology a. Population growth is another example of the derivative used in science Suppose n=f(t) is the number of individuals in some plants and animals populations as the time t the change in the populations size time t1 and t2 n = f(t2) – f(t1)
- 8. In chemistry A .One use of derivatives in chemistry is when you want to find out the concentrations Of an element in a priducts b. Derivative is used to find the rate of reaction and compressability in chemistry And many more uses in chemistry that are beyond the approach at basic level In mathematics Is used to find; Extreme value of functions The means value theorms Monotonic functions Curves sketching Newton and methods etc.
- 9. In other sciences a) Rate of heat flow in geology. b) Rate of spread of roumors in sociology c) Rate of improvement of performance in psychology d) Inshort ,, Derivatives is widely used in every field of life and in every field of science almost , to find how much the rate of change of particular process or function.
- 10. THANK YOU