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Summary Mathematics chapter Differenial equation

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The chapter on differential equations is a core topic in mathematics that deals with equations that describe how a system changes over time. Differential equations are used to model a wide range of phenomena, from the motion of objects to the spread of disease and the behavior of economic systems. This chapter introduces students to the basic concepts of differential equations, including the various types of differential equations, their properties, and their solutions. Students will learn how to classify differential equations, and how to apply different techniques to solve them, including separation of variables, linear equations, and systems of equations. One of the key features of this chapter is the emphasis on real-world applications of differential equations. Students will learn how to use differential equations to model a variety of physical and biological systems, and how to interpret the results of their solutions in terms of the original problem. In addition to the mathematical principles of differential equations, this chapter also explores the historical and cultural significance of the concept. Students will learn about the contributions of famous mathematicians such as Euler and Laplace to the development of differential equations, as well as the role of differential equations in the history of science and technology. Overall, this chapter provides a comprehensive and engaging introduction to the topic of differential equations, with plenty of examples and exercises to help students master the material. By the end of the chapter, students will have a deep understanding of the principles of differential equations and how they can be used to model and solve real-world problems.

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Definition
An gin involving one dependent variable
more
and its derivatives w.lt one


independent variably is Differential Equatio



g
41 5
4 65inn 32


22 5
N
Ent y Ey In

It is dependent on my

Types
n
of D E
n



4 Ordinary D E

A D E involving derivatives w.lt singl
independent variably is ordinary D E


Elt zag using 2En
Cii Partial D E
A D E involving at least two independe

, variably and partial derivatives with
respect to either of these variables
is called a Partial D t

34210g ayy
CI's x
Ey sing
Ey
Order and Degree of Dot

I Order's is the order of highest
order derivative

eat dog dy 2sina 3y

Order 3
I Degree The exponent of the highest
order derivative of simplified as
polynomial form of given DoE
make free from radical sign

etc dy IIs anta 3

, order 5 degree 2


i Sin Ez 322

sin 322
d4
order L degree 1

day
iii sin
Ies 319 87
Ordaz 2
is not defined
degree
since given D E Can't
simplified to polynomial
form

in dg a

3 s
Eg a dt
order 2 Degree Not defined

, a f to
Order 3 degree 2


ai n it Eat 4474347
Han
n e


Gu log

order I degree I

Ein
Formation of Diff
Let given c in
n 0
f x y Cr Ca
Ci G i Cn are arbitrary
constants

Differentiating the given Ein n
times W e t a we get n more
gins
By eliminating the arbitrary
constants a Cai ien weset

Document information

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Uploaded on
March 30, 2023
Number of pages
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Written in
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Type
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