1. Shantilal Shah Engineering College

2. Advance Engineering Maths (2130002)

3. Group Members Guided By : S. J. VORA
Senjaliya kevin ( ) ;
Pargi vishal ( ) ;
Timbadiya akash ( ) ;
Pandya nandish ( ) ;
Trivedi niraj ( ) ;
Guided By : S. J. VORA

4. (1) Ordinary Differential Equation and Application

5. Differential Equation
A differential equation is an equation containing derivatives or differentials of one or more dependent variables with respect to one or more independent variables.

6. Differential Equation
Here is an example of differential equations.
It is the differential equation governing the linear displacement x(t) of a body of mass m, subjected to an applied force F(t) and a restraining spring of stiffness k.

7. A differential equation has the form Its general solution is
Differential Equations
A differential equation has the form
Its general solution is

8. Ex. Find the general solution to
Differential Equations
Ex. Find the general solution to

9. Differential Equations
Find the general solution to

10. Ordinary and Partial Differential Equation
A differential equation containing a single independent variable and the derivatives with respect to it is called an ordinary differential equation.
e.g.
This equation is an ordinary differential equation.

11. Ordinary and Partial Differential Equation
A differential equation containing more than one independent variables and the partial derivatives with respect to them is called partial differential equation.
Ex.

12. Classification of ordinary differential equation
The order of an ordinary differential equation is the order of the highest derivative involved in the equation.
The degree of an ordinary differential equation is the degree of the highest derivative, which occurs in it when the differential coefficients have been made free from radicals and factions, if any.

13. Differential Equation

14. Linear and nonlinear homogeneous equations
A differential equation is said to be linear if the dependent variable and every derivatives in the equation occurs in the first degree only and they should not be multiplied together.
Ex.

15. Linear and nonlinear homogeneous equations
When the input function f(x) = 0 then given equation reduces to the associate homogeneous equation.
Otherwise is known as nonhomogeneous equation.

16. Linear and nonlinear homogeneous equations
When one or more of the coefficients of the above equation depends on x, it is called an equation with variable coefficients.
Otherwise it will become a linear equation with constant coefficients.

17. Linear and nonlinear homogeneous equations
An ordinary differential equation that is non linear is called nonlinear ordinary differential equation.

18. Solution of ordinary differential equation
A solution of an nth ordinary differential equation that contains n arbitrary constants is called the general solution of differential equation.

19. Solution of ordinary differential equation
A solution obtained from a general solution by giving particular values to one or more arbitrary constants is called particular solution of an ordinary differential equation.
However the general solution is always unique if it exist. Again the solution of an differential equation, if it exist, is always a continuous function.

20. Solution of ordinary differential equation
A solution of an ordinary differential equation that cannot be obtained from the general solution for any choice of its arbitrary constants is called singular solution.

21. TEAM WORK

22. (2) Series Solution of Differential Equation

23. Power Series An infinite series of the form :
Is called a power series in (x – x0).
If we put x0 = 0

24. Analytic Function
A function f(x) defined on a interval containing the point x = x0 is called analytic at x0.
A function f(x) defined on a interval containing the point x=x0 is called non-analytic at x0.

25. Ordinary and Singular Point
Consider the linear differential equation
y” + P(x)y’ + Q(x)y = 0
where p(x) and q(x) are functions of x only.

26. Ordinary and Singular Point
1. The point x = x0 is called an ordinary point of (1) if both the functions p(x) and q(x) are analytic at x = x0
If the point x = x0 is not an ordinary point then it is called a singular point.
(i) Regular singular point : if both (x - x0)p(x) and (x - x0)2q(x) are
analytic at x = x0.
(ii) Irregular singular point : if the point x = x0 is not regular it is called irregular singular point.

27. Taylor Series
Suppose that  an(x - x0)n converges to f (x) for |x - x0| < .
Then the value of an is given by
and the series is called the Taylor series for f about x = x0.
Also, if
then f is continuous and has derivatives of all orders on the interval of convergence. Further, the derivatives of f can be computed by differentiating the relevant series term by term.

28. Bessel’s equation Bessel’s differential equation is written as
or in standard form,

29. General solution of Bessel’s equation
Bessel’s function of the first kind of order n
is given as,
Substituting –n in place of n, we get

30. General solution of Bessel’s equation
If n is not an integer, a general solution of Bessel’s equation for all x≠0 is given as

31. Properties of Bessel’s Function
J0(0) = 1
Jn(x) = 0 (n>0)
J-n(x) = (-1)n Jn(x)

32. LEGENDRE POLYNOMIALS n = 0 P0(x) = 1 n = 1 P1(x) = x n = 2
P3(x) = ½ (5x3-3x)
n = 4
P4(x) = 1/8 (35x4-30x2+3)
n = 5
P5(x) = 1/8 (63x5-70x3+15x)
n = 6
P6(x) = 1/16 (231x6-315x4+105x2-5)

33. TEAM WORK

34. THANK YOU