Shantilal Shah Engineering College

Advance Engineering Maths (2130002)

Group Members Guided By : S. J. VORA Senjaliya kevin (130430109048) ;9727875091 Pargi vishal (130430109037) ;9687786998 Timbadiya akash (130430109054) ;8469449860 Pandya nandish (130430109034) ;7405656180 Trivedi niraj (130430109056) ;7359311725 Guided By : S. J. VORA

(1) Ordinary Differential Equation and Application

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.

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.

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

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

Differential Equations Find the general solution to

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.

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.

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.

Differential Equation

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.

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.

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.

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

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.

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.

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.

TEAM WORK

(2) Series Solution of Differential Equation

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

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.

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.

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.

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.

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

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

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

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

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)

TEAM WORK

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