© 2019 JETIR January 2019, Volume 6, Issue 1 Computation ...

© 2019 JETIR January 2019, Volume 6, Issue 1 www.jetir.org (ISSN-2349-5162)

JETIRDY06137 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 837

Computation of Some Special Ordinary Differential

Equations (Chebyshev & Clairaut)

A.K. Awasthi*,Vijay Kumar Tripathi

Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University,

Punjab, India*

Directorate of Distance Education, Guru Jambheshwar University of Science and Technology, Hisar, Haryana,

India

1. Abstract

This paper is comprising of the solution of ordinary differential equations using MATHEMATICA software.

Some special ordinary differential equations namely Chebyshev differential equation and Clairaut differential

equation are solved by MATHEMATICA software.

2. Introduction

An Ordinary differential equation (ODE) is an equation of the function and involves derivatives of the function.

ODE can be written as “diff eq.” and “diffy Q”. The form of 𝑛𝑡ℎ order ordinary differential equation is,

(1)

In eq. [1] y is a function of x and 𝑦′ is the first derivative with respect to x, it can be written in the form

Similarly, 𝑦(𝑛) is the 𝑛𝑡ℎ derivative with respect to x, it can be written as,

Non-homogeneous ordinary differential equations can be solved if the general solution to the homogenous

version is known, in which case the undetermined coefficients methods or variation of parameters can be used to

find the particular solution.

In [1] Awasthi discussed the ordinary differential equations and use Mathematica software for the solve

equations. Anger and Baer differential equations solved use of Mathematica software by Awasthi[2]. In [3]

Conceicao et al. discussed about some mathematical F- Tools, such as F- Quardratic, F- Linear, F- Trignometric,

etc, by Mathematica software. Garrppa [4] discussed the solution of the nonlinear systems involved in implicit

methods and also discussed of fractional- order problems.

3. Basics of commands for solving Ordinary Differential Equations

ODE can be solved by Wolfram language commands in MATHEMATICA software. i.e.

DSolve [eqn, y, x]

For numerically used command i.e.

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3.1. Classification and background of 𝒏𝒕𝒉 -order Ordinary Differential Equations

n-order ODE is said to be linear and form of n- order ODE i.e.

(2)

where Q(x) = 0 is said to be homogeneous. Confusingly, form of an ODE

Some-times i.e. called “Homogeneous”.

Normally, in nth order Ordinary differential equation has n linearly independent solutions. While, solution of

linearly independent function have any linear combination then it is also a solution.

Ordinary differential equations have first order as integrating factor, and second order as Sturm-Liouville theory,

ordinary differential equations and can be solved arbitrary ordinary differential equations with linear constant

coefficients, when it is present in a certain factorable form. Laplace transform is a form of integral transform, can

be use to solve classes of linear ordinary differential equations.

There are many techniques for solve the classes of ordinary differential equations. For solve the complicated

equations have only particular solution, Numerical methods. There are many methods are developed such as the

collection method, Galerkin method, Runge-Kutta method. Runge-Kutta method is most popular in these

methods. In research and number of publications has devoted to the numerical solution of partial and ordinary

both differential equations as result of their importance in fields as diverse as engineering, physics, electronics

and economics.

The solution of an ODE satisfies two properties. i.e. Existence and Uniqueness. By Picard's existence theorem

can be establish these properties for certain classes of ordinary differential equations. First order ordinary

differential equation as,

(3)

In eq. (3) i = 1,2,3,…,n and let the function 𝑓𝑖(𝑥1, 𝑥2, … , 𝑥𝑛, 𝑡), where i = 1,2,3,…,n, all be defined in (n+1)-

dimensional space domain D of the variables 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑡. Let these functions be continuous in domain D. In

domain D have continuous first Partial derivatives 𝜕𝑓𝑖

𝜕𝑥𝑗 for i = 1,2, 3,…,n and j = 1,2,3,…,n. Suppose

(𝑥10, 𝑥2

0, 𝑥30 … , 𝑥𝑛

0) be in domain D. Then there exists a solution of eq. (3). i.e.

(4)

For (where ) satisfying the initial conditions

(5)

Further, the solution is unique, so that if




(6)

is a second solution of (1) for satisfying (1),

then for . Since every nth order ordinary differential equations can be expressed as a

system of n first- order ODE. This theorem apply for the single nth-order ODE.

3.2. Basics of exact first order Ordinary Differential Equations

Form of an exact first order ordinary differential equation is,

(7)

Where,

(8)

An equation of the form (1) with

(9)

is said to be non-exact. If

(10)

in (1), it has an x -dependent integrating factor. If

(11)

in (1), it has an x y -dependent integrating factor. If

(12)

in (1), it has a y-dependent integrating factor.

Other special first order types include cross multiple equations

Name of the first order differential equations Equations

Homogeneous equations

Linear equations Separable equations

3.3. Basics of Special classes of Second Ordinary Differential Equations

Special classes of the second order differential equations include,




(13)

If x is not available in equation then,

(14)

(If y is not available in equation). A second-order linear homogeneous ODE

(15)

for which can be transformed to one with constant coefficients.i.e.

(16)

The equation of simple harmonic motion in undamped form,

(17)

which becomes

(18)

when damped, and

(19)

when both forced and damped.

Form of the Systems with constant coefficient. i.e.

(20)

Some common and important examples of ordinary differential equations arise in problems of mathematical

physics.

4. Chebyshev differential equation and Solution by MATHEMATICA Software

Form of the Chebyshev differential equation. i.e.

(21)

Chebyshev differential equation can solve by MATHEMATICA Software.

Programming for the solution of Chebyshev differential equation.

Programming for Chebyshev differential equation




5. Clairaut differential equation and Solution by MATHEMATICA Software

Form of the Clairaut differential equation

(22)

Clairaut differential equation can solve by MATHEMATICA Software.

Programming for the solution of Clairaut differential equation.

Programming for Clairaut differential equation




6. Conclusion

Some special ordinary differential equations Chebyshev differential equation and Clairaut differential equation

can be solve by above Programming through MATHEMATICA software.

7. References

1. Conceição, A., Pereira, J., Silva, C., & Simão, C. (2012). Mathematica in the classroom: new tools for

exploring precalculus and differential calculus. In

Proceedings of the 1st National Conference on Symbolic

Computation in Education and Research.

2. Garrappa, R. (2018). Numerical solution of fractional differential equations: A survey and a software

tutorial.

Mathematics,

6(2), 16.


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