Presentations Differential equation.pptx

1. Differential Equation
Mr. Atma Nand, Assistant Professor
School of Applied and Life Sciences (SALS)
Uttaranchal University, Dehradun

2. • INTRODUCTION TO DIFFERENTIAL EQUATION
• ORDER AND DEGREE OF A DIFFERENTIAL EQUATION
• CLASSIFICATIONS OF DIFFERENTIAL EQUATION
• FORMATION OF DIFFERATIAL EQUATION
• SOLUTION OF DIFFERENTIAL EQUATION
• BASIC CONCEPT OF PDE (PARTIAL DIFFERENTIAL EQUATION)
• APPLICATIONS OF DIFFERENTIAL EQUATIONS
Learning Outcomes:
DIFFERENTIAL EQUATIONS

3. DIFFERENTIAL EQUATIONS
• An equation involving the Independent Variable x,
dependent Variable y and the differential coefficients of
dependent Variable with respect to independent
variable is called a Differential Equation.
• An equation involving one or more partial derivatives of
an unknown function of two or more independent
variables is known as a partial differential
equation.

4. For Example:

5. TYPES OF DIFFERENTIALEQUATION
ODE (ORDINARY DIFFERENTIAL EQUATION):
An equation contains only ordinary derivates of one or more
dependent variables of a single independent variable.
For Example,
dy/dx + 5y = ex, (dx/dt) + (dy/dt) = 2x + y
PDE (PARTIAL DIFFERENTIAL EQUATION):
An equation contains partial derivates of one or more
dependent variables of two or more independent variables.
For Example,

6. Order of a Differential Equation
• The Order of a Differential equation is the order of the
highest derivative occurring in the Differential equation
𝑑3
𝑦 𝑑2
𝑦 2
_ 𝑑𝑦
• Eg : (i) + 2 = 0
𝑑𝑥3
𝑑𝑥2
𝑑𝑥
Order of the equation is 3
(ii)
𝑑2
𝑦
= 1 +
𝑑𝑦
𝑑𝑥2
𝑑𝑥
Order of the equation is 2

7. dx
dy
 2x  3
3  9y  0
dx2
dx
d 2
y

dy

   6y  3
dx3
 dx
d 3
y  dy
4
ORDER
2
3
1
DIFFERETIAL EQUATION

8. Degree of a Differential Equation
• The Degree of a Differential equation is the degree of
the highest derivative occurring in the Differential
equation
𝑑3
𝑦 𝑑2
𝑦 2
_ 𝑑𝑦
• Eg : (i) + 2 = 0
𝑑𝑥3
𝑑𝑥2
𝑑𝑥
Degree of the equation is 1
(ii) = 1 +
𝑑2
𝑦 𝑑𝑦
𝑑𝑥2
𝑑𝑥
Degree of the equation is 2
• Note: Order and degree (if defined) of a differential
equation are always positive integers.

9. Formation of ordinary differential
equation
Consider the equation f ( x, y ,c1 ) = 0 -------(1) where c1 is the
arbitrary constant. We form the differential equation from this
equation. For this, differentiate equation (1) with respect to the
independent variable occur in the equation.
Eliminate the arbitrary constant c from (1) and its derivative.
Then we get the required differential equation.
Suppose we have f ( x, y ,c1 ,c2 ) = 0 . Here we have two arbitrary
constants c1 and c2 . So, find the first two successive derivatives.
Eliminate c1 and c2 from the given function and the successive
derivatives. We get the required differential equation.
Note: The order of the differential equation to be formed is equal to the number
of arbitrary constants present in the equation of the family of curves.

10. Example1: Find the differential equation of the family of straight lines
y=mx+c when (i) m is the arbitrary constant (ii) c is the arbitrary
constant (iii) m and c both are arbitrary constants.
Solution:

13. Example2: Find the differential equation of the family of curves
y= a/x + b where a and b are arbitrary constants.
Solution:

14. Classifications of Differential Equation
• Classifications of Differential Equation
depends on their
(i) Order
(ii) Linearity

15. Classifications of Differential
Equation according to
their Order
• First Order Differential Equation
First Order Differential Equation are those in
which only the First Order derivative of the dependent
variable occurs.
• Higher Order Differential Equation
Differential equations of order two or more are
referred as higher Order Differential Equation.

16. TYPES OF ODE
FIRST ORDER ODE
• FIRST ORDER LINEAR ODE
• EXACT EQUATION
• NON-LINEAR FIRST ORDER ODE
• SEPERABLE EQUATION
• BERNOULLI DIFFERENTIAL EQUATION
SECOND ORDER ODE
• LINEAR SECOND ORDER ODE
• HOMOGENEOUS SECOND ORDER ODE
• INITIAL AND BOUNDARY VALUE PROBLEMS
• NON-LINEAR SECOND ORDER ODE
• NON-HOMOGENEOUS SECOND ORDER ODE
HIGHER ORDER ODE
• LINEAR NTH ORDER ODE
• HOMOGENEOUS EQUATION
• NON-HOMOGENEOUS EQUATION

17. FIRST ORDER ODE
• A first order differential equation is an
equation involving the unknown function y,
its
derivative y' and the variable x. We will only talk
about explicit differential equations.
• General Form,
𝑑𝑦
𝑑𝑥
= 𝐹(𝑥, 𝑦)
• For Example, dx
dy
 2x  3

18. LINEAR DIFFERENTIAL EQUATIONS
• A differential equation of the form
𝑑y
𝑑x
+ Py = Q, where P and Q are constants or
functions of x only is called a first order linear
differential equation.

19. FIRST ORDER linear ODE
• A first order linear differential equation
has the following form:
𝑑𝑦
𝑑𝑥
+ 𝑝 𝑥 𝑦 = 𝑞(𝑥)
• The general solution is given by
𝑦 × 𝐼. 𝐹. = 𝑦 𝑥 𝑞 𝑥 𝑑𝑥 + 𝐶
Where 𝐼. 𝐹. = 𝑒 𝑝 𝑥 𝑑𝑥
called the integrating factor. If an initial condition is given,
use it to find the constant C.

20. EXACT EQUATION
• Let a first order ordinary differential
equation be expressible in this form
M(x,y)+N(x,y)dy/dx=0
such that M and N are not homogeneous functions
of the same degree.
• However, suppose there happens to exist
a function f(x,y) such that:
∂f/∂x=M, ∂f/∂y=N
such that the second partial derivatives of f exist and
are continuous.
• Then the expression Mdx+Ndy is called an exact
differential, and the differential equation is called an
exact differential equation.

21. DIFFERENTIAL EQUATIONS WITH VARIABLES
SEPARABLE
• If F (x, y) can be expressed as a product g (x) and h(y),
where, g(x) is a function of x and h(y) is a function of
𝑑y
𝑑x
= F(x,y) is said to
y, then the differential equation
be of variable separable type.

22. SEPERABLEQUATION
• A separable differential equation is any differential
equation that we can write in the following form.
• Note that in order for a differential equation to be
separable all the y's in the differential equation must be
multiplied by the derivative and all the x's in the
differential equation must be on the other side of the
equal sign.

23. BERNOULLI
EQUATION:
• where p(x) and q(x) are continuous functions on the interval we’re working
on and n is a real number. Differential equations in this form are called Bernoulli
Equations.
• First notice that if 𝑛 = 0 or 𝑛 = 1 then the equation is linear and we already know how to solve it
in these cases. Therefore, in this section we’re going to be looking at solutions for values of n
other than these two.
• In order to solve these we’ll first divide the differential equation by 𝑦𝑛 to
get,
• We are now going to use the substitution𝑣 = 𝑦1−𝑛 to convert this into a differential equation in
terms of v. As we’ll see this will lead to a differential equation that we can solve.

24. SECOND ORDER
ODE:
• The most general linear second order differential equation is in
the form.
• In fact, we will rarely look at non-constant coefficient linear
second order differential equations. In the case where we
assume constant coefficients we will use the following
differential equation.
• Initially we will make our life easier by looking at differential
equations with g(t) = 0. When g(t) = 0 we call the Differential
Equation Homogeneous and
When g(t) ≠ 0 we call the Differential Equation Non-
Homogeneous.

25. HIGHER ORDER
ODE
• The most general nth order linear differential
equation is, nth – order linear differential equation
with constant coefficients
a0 y  gx
d 2
y

dy

a
dx2 1
dx
dxn1
d n1
y
 ....  a2
dxn
d n
y
 an1
an
• nth – order linear differential equation
with variable coefficients,
dxn
x x y  g x
dx
dy
an x  x
dx
dn1
y
an1
d 2
y dy
 ......  a2 x  a1   a0   
dx2

26. • For Example,
• The above equation is an example of Higher
Order Homogeneous Differential ODE with
initial conditions.
• Similarly, the above equation is an Higher
Order Non-Homogeneous Differential ODE
with coefficients.

27. Solution of a differential equation
A function which satisfies the given differential
equation is called its solution or integral.

28. GENERAL AND PARTICULAR SOLUTIONS OF A
DIFFERENTIAL EQUATION
• The solution which contains arbitrary constants is called
the
general solution (primitive) of the differential
equation.
• The solution free from arbitrary constants i.e., the
solution obtained from the general solution by giving
particular values to the arbitrary constants is called a
particular solution of the differential equation.

29. Examples of Differential Equations and Their Solutions
Note that a solution to a differential equation is not necessarily unique, primarily
because the derivative of a constant is zero. For example, 𝑦 = 𝑥2
+ 4 is also a
solution to the first differential equation in table

31. APPLICATIONS OF
ODE
MODELLING WITH FIRST-ORDER EQUATIONS
 Newton’s Law of Cooling
 Electrical Circuits
MODELLING FREE MECHANICAL OSCILLATIONS
 No Damping
 Light Damping
 Heavy Damping
MODELLING FORCED MECHANICAL OSCILLATIONS
COMPUTER EXERCISE OR ACTIVITY