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1. NAME: ISHANI MAZUMDAR
DEPARTMENT: CSE
SEMESTER: 3RD SEMESTER
YEAR: SECOND YEAR
ROLL NO: 10800121114
BATCH:B
TOPIC:DIFFRENTIAL
EQUATIONS

2. INDEX
2
1. INTRODUCTION
2. TYPES OF DIFFERENTIAL EQUATIONS
3. ORDINARY DIFFERENTIAL EQUATIONS
4. PARTIAL DIFFERENTIAL EQUATIONS
5. NON-LINEAR DIFFERENTIAL EQUATIONS
6. REAL-LIFE PROBLEMS
7. RULES
8. EXACT EQUATIONS
9. SOLVE
10.CONCLUSIONS
11.BIBLIOGRAPHY

3. DIFFERENTIAL EQUATIONS
3
 In mathematics, a differential equation is an equation that relates one or more unknown functions and
their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their
rates of change, and the differential equation defines a relationship between the two. Such relations are common;
therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics,
and biology.
 Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically
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.
 Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy
each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by
explicit formulas; however, many properties of solutions of a given differential equation may be determined without
computing them exactly.

4. Ordinary differential
equations
Partial differential
equations
Non-linear differential
equations
TYPES OF DIFFERENTIAL
EQUATIONS
Differential equations can be divided into
several types. Apart from describing the
properties of the equation itself, these
classes of differential equations can help
inform the choice of approach to a
solution. Commonly used distinctions
include whether the equation is ordinary
or partial, linear or non-linear, and
homogeneous or heterogeneous. This list
is far from exhaustive; there are many
other properties and subclasses of
differential equations which can be very
useful in specific contexts.

5. Ordinary differential equations
 An ordinary differential equation (ODE) is an equation containing an unkAn ordinary differential equation (ODE) is an
equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x.
The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Thus x is
often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial
differential equation, which may be with respect to more than one independent variable.
 Linear differential equations are the differential equations that are linear in the unknown function and its derivatives.
Their theory is well developed, and in many cases one may express their solutions in terms of integrals.
 Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of
linear differential equations (see Holonomic function).
 As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical
methods are commonly used for solving differential equations on a computer.nown function of one real or complex
variable x, its derivatives, and some given functions of x. The unknown function is generally represented by
a variable (often denoted y), which, therefore, depends on x. Thus x is often called the independent variable of the
equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect
to more than one independent variable.
 Linear differential equations are the differential equations that are linear in the unknown function and its derivatives.
Their theory is well developed, and in many cases one may express their solutions in terms of integrals.
 Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of
linear differential equations (see Holonomic function).
 As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical
methods are commonly used for solving differential equations on a computer. 5

6. Partial differential equations
d drink breaks
If you need to use the restroom, get a drink, or see the nurse, please ask permission and then
record A partial differential equation (PDE) is a differential equation that contains
unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary
differential equations, which deal with functions of a single variable and their derivatives.) PDEs
are used to formulate problems involving functions of several variables, and are either solved in
closed form, or used to create a relevant computer model.
PDEs can be used to describe a wide variety of phenomena in nature such
as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics.
These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just
as ordinary differential equations often model one-dimensional dynamical systems, partial
differential equations often model multidimensional systems. Stochastic partial differential
equations generalize partial differential equations for modeling randomness.
 on the sheet posted on the classroom door.
6

7. Non-linear differential equations
 A n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n i s a d i f f e r e n t i a l e q u a t i o n t h a t i s n o t
a l i n e a r e q u a t i o n i n t h e u n k n o w n f u n c t i o n a n d i t s d e r i v a t i v e s ( t h e l i n e a r i t y
o r n o n - l i n e a r i t y i n t h e a r g u m e n t s o f t h e f u n c t i o n a r e n o t c o n s i d e r e d h e r e ) .
T h e r e a r e v e r y f e w m e t h o d s o f s o l v i n g n o n l i n e a r d i ff e r e n t i a l e q u a t i o n s
e x a c t l y ; t h o s e t h a t a r e k n o w n t y p i c a l l y d e p e n d o n t h e e q u a t i o n h a v i n g
p a r t i c u l a r s y m m e t r i e s . N o n l i n e a r d i f f e r e n t i a l e q u a t i o n s c a n e x h i b i t v e r y
c o m p l i c a t e d b e h a v i o u r o v e r e x t e n d e d t i m e i n t e r v a l s , c h a r a c t e r i s t i c
o f c h a o s . E v e n t h e f u n d a m e n t a l q u e s t i o n s o f e x i s t e n c e , u n i q u e n e s s , a n d
e x t e n d a b i l i t y o f s o l u t i o n s f o r n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s , a n d w e l l -
p o s e d n e s s o f i n i t i a l a n d b o u n d a r y v a l u e p r o b l e m s f o r n o n l i n e a r P D E s a r e
h a r d p r o b l e m s a n d t h e i r r e s o l u t i o n i n s p e c i a l c a s e s i s c o n s i d e r e d t o b e a
s i g n i f i c a n t a d v a n c e i n t h e m a t h e m a t i c a l t h e o r y ( c f . N a v i e r – S t o k e s
e x i s t e n c e a n d s m o o t h n e s s ) . H o w e v e r, i f t h e d i f f e r e n t i a l e q u a t i o n i s a
c o r r e c t l y f o r m u l a t e d r e p r e s e n t a t i o n o f a m e a n i n g f u l p h y s i c a l p r o c e s s , t h e n
o n e e x p e c t s i t t o h a v e a s o l u t i o n . [ 11 ]
 L i n e a r d i f f e r e n t i a l e q u a t i o n s f r e q u e n t l y a p p e a r a s a p p r o x i m a t i o n s t o
n o n l i n e a r e q u a t i o n s . T h e s e a p p r o x i m a t i o n s a r e o n l y v a l i d u n d e r r e s t r i c t e d
c o n d i t i o n s . F o r e x a m p l e , t h e h a r m o n i c o s c i l l a t o r e q u a t i o n i s a n
a p p r o x i m a t i o n t o t h e n o n l i n e a r p e n d u l u m e q u a t i o n t h a t i s v a l i d f o r s m a l l
a m p l i t u d e o s c i l l a t i o n s .

8. 8
ORDINARY DIFFERENTIAL EQUATIONS
dy/dx+7y=8
Here,
x= independent term(input)
y= dependent term(output)
PARTIAL DIFFERENTIAL EQUATIONS
∂u/∂x+∂u/∂y=2x+3y
u has to be function of both x and y

9. Real-life problems
In a sudden place , police were called at about 3pm , where a murder victim was
found. After coming to place , police look temperature of a deceased body which
was 34.5°C.
After one hour, police again took the reading of temperature of body which was found to
be 33.5°C. The temperature of the room was 15° then what is the murder time?
To solve the problem we need to know , the Newton’s Cooling Principle, which states that
the rate of cooling of a body is proportional to the difference between its temperature
and the temperature of the surrounding air.

10. Continue.....e and be respectful?
TIME(t) temperature(d)
t=0 34.5°
t=1hr 33.9°
dQ/dt α (Q-15)
dQ/dt = k(Q-15)
dQ/(Q-15) = k dt
Integrating both sides
ꭍ dQ/(Q-15) = k ꭍ dt
ln|Q-15|= kt + c
Put t=0
ln|34.5-15|=c
c= 19.5
Put t=1,Q=33.9
Ln|33.9-15|=k.1+ln 19.5
K=-0.3125
The particular solution is
Ln|Q-15|=-0.3125t + ln 19.5
Ln|37.15|=-0.3125t + ln19.5 , when Q=37°C
t= -0.386 hr
= -0.386*60min
=-23min

11. Rules for finding out integrating factor :-

12. Rule 1 :
If (∂M/∂y - ∂N/∂x)/N = f(x) only , then IF= e^ꭍf(x)dx
Rule 2:
If (∂M/∂y - ∂N/∂x)/M = g(y) only , then IF=e^{-ꭍg(y)}dy
Rule 3:
If differential equations is in the form of M dx+ N dy =0; and also M and N are homogeneous functions in x , y;
then
IF=1/(Mx + Ny)
Rule 4:
If the equations M dx + N dy=0 , can be represented in the form of
F(x + y)dx= x g(xy)dy =0 den ,
IF= 1/(Mx + Ny)
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13. EXACT EQUATIONS
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14. 14
(x^2y−2xy^2)dx=(x^3−3x^2y)dy
(x^2y−2xy^2)dx−(x^3−3x^2y)dy=0
Here,
M=x^2y−2xy^2⇒∂y/∂M​=x^2−4xy
N=−(x^3−3x^2y)⇒∂x/∂N​=6xy−3x^2
∵∂y/∂M​=∂x/∂N​
Therefore,
Integrating factor =1/Mx + Ny​=1/(x(x^2y−2xy^2)+y(3x^2y−x^3))=1/x^2y^2
Multiplying the given equation by the integrating factor, we get
1/(x^2y^2​)(x^2y−2xy^2)dx−1/(x^2y^2)(x^3−3x^2y) dy=0
⇒(1/y−2/x​)dx−(x/y^2−3/y​) dy=0
Now again,
M=(1/y​−2/x​)⇒∂y/∂M​=−1/y^2
N=−(x/y^2−3/y)⇒∂x/∂N​=−1/y^2​
Now the above equation is an exact differential equation.
Therefore,
Solution of the equation is-
∫Mdx+∫(terms in N not containing x)dy=C
⇒∫(1/y​−2/x​)dx+∫(3/y​)dy=C
⇒y/x​−2logx+3logy=C
⇒y/x​−logx2+logy3=C
⇒y/x​+log(x^2y^3​)=C
Solve : (x^2-2xy^2)dx+(3x^2-x^3)dy

15. CONCLUSION
Differential equations plays major role in applications of sciences and engineering. It arises in
wide variety of engineering applications for e.g. electromagnetic theory, signal processing,
computational fluid dynamics, etc. These equations can be typically solved using either analytical
or numerical methods. Since many of the differential equations arising in real life application
cannot be solved analytically or we can say that their analytical solution does not exist.
It has been pointed out that the employment of neural network architecture adds many
attractive features towards the problem compared to the other existing methods in the
literature. Preparation of input data, robustness of methods and the high accuracy of the
solutions made these methods highly acceptable. The main advantage of the proposed approach
is that once the network is trained, it allows evaluation of the solution at any desired number of
points instantaneously with spending negligible computing time. Moreover, different hybrid
approaches are also available and the work is in progress to use better optimization algorithms.
People are also working in the combination of neural networks to other existing methods to
propose a new method for construction of a better trail solution for all kind of boundary value
problems. Such a collection could not be exhaustive; indeed, we can hope to give only an
indication of what is possible.
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16. I would like to thank my teacher, Dr. Animesh Upadhyay who gave me this
opportunity to work on this project. I got to learn a lot from this project about
DIFFERENTIAL EQUATIONS. I would also like to thank our college principal.
At last, I would like to extend my heartfelt thanks to my parents because without
their help this project would not have been successful. Finally, I would like to
thank my dear friends who have been with me all the time.
16
Acknowledgement

17. Bibliography
1. Wikipedia
2. Google
3. Das and Paul maths book

18. 18