Computation of Bernoulli and Bessel Differential Equation

IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 405

Computation of Bernoulli and Bessel Differential

Equation A.K. Awasthi1, Arun Kumar Garov2, Sanjeev Kumar3

Professor, Department of Mathematics, Lovely Professional University, Punjab, India1

Research Scholar, Department of Mathematics, Lovely Professional University, Punjab, India2,3

Abstract

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

ordinary differential equations namely Bernoulli differential equation and Bessel differentia equation are solved by

MATHEMATICA software.

1.1 Basics of Ordinary Differential Equations

An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a

function and its derivatives An ODE of order n is an equation of the form

𝐹(𝑥, 𝑦, 𝑦 ′, … , 𝑦(𝑛)) = 0 (1)

Where y is a function of x and 𝑦 ′ is first derivative with respect to x, it can be written in the form

𝑦 ′ =𝑑𝑦

𝑑𝑥

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

𝑦(𝑛) =𝑑𝑛𝑦

𝑑𝑥𝑛

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.

1.2 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.

NDSolve [eqn, y, {x, xmin, xmax}]

1.3 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.

𝑎𝑛(𝑥)𝑦(𝑛) + 𝑎𝑛−1(𝑥)𝑦(𝑛−1) + ⋯ + 𝑎1(𝑥)𝑦 ′ + 𝑎0(𝑥)𝑦 = 𝑄(𝑥) (2)

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

𝑦 ′ =𝑑𝑦

𝑑𝑥

is also some-times called "homogeneous”.

In general, an nth-order ODE has n linearly independent solutions. Furthermore, any linear combination of linearly

independent function solutions is also a solution.

Simple theories exist for first- order (integrating factor) and second order (Sturm-Liouville theory) ordinary

differential equations, and arbitrary ODEs with linear constant coefficients can be solved when they are of certain

factorable forms. Integral transforms such as the Laplace transform can also be used to solve classes of linear

While there are many general techniques for analytically solving classes of ODEs, the only practical solution

technique for complicated equations is to use numerical methods (Milne 1970, Jeffreys and Jeffreys 1988). The

most popular of these is the Runge-Kutta method, but many others have been developed, including the collocation

method and Galerkin method. A vast amount of research and huge numbers of publications have been devoted to

the numerical solution of differential equations, both ordinary and partial (PDEs) as a result of their importance in

fields as diverse as physics, engineering, economics, and electronics.

The solutions to an ODE satisfy existence and uniqueness properties. These can be formally established by

Picard’s existence theorem for certain classes of ODEs. Let a system of first order ODE be given by

𝑑𝑥𝑖

𝑑𝑡= 𝑓𝑖(𝑥1, 𝑥2, … , 𝑥𝑛, 𝑡) (3)

For i=1,2,3,…,n and let the functions 𝑓𝑖(𝑥1, 𝑥2, … , 𝑥𝑛, 𝑡), where i=1,2,3,…,n, all be defined in a domain D of the

(n+1)-dimensional space of the variables 𝑥1, 𝑥2, … , 𝑥𝑛, 𝑡. Let these functions be continuous in D and have

continuous first Partial derivatives 𝜕𝑓𝑖

𝜕𝑥𝑗 for i=1,2,3,…,n and j=1,2,3,…,n in D. Let(𝑥1

0, 𝑥20, 𝑥3

0 … , 𝑥𝑛0) be in D. Then

there exists a solution of (3) given by

𝑥1 = 𝑥1(𝑡), 𝑥2 = 𝑥2(𝑡), … , 𝑥𝑛 = 𝑥𝑛(𝑡) (4)

For 𝑡0 − 𝛿 < 𝑡 < 𝑡0 + 𝛿 (where > 0 ) satisfying the initial conditions

𝑥1(𝑡0) = 𝑥10, 𝑥2(𝑡0) = 𝑥2

0, … , 𝑥𝑛(𝑡0) = 𝑥𝑛0 (5)

Furthermore, the solution is unique, so that if

𝑥1 = 𝑥1∗(𝑡), 𝑥2 = 𝑥2

∗(𝑡), … , 𝑥𝑛 = 𝑥𝑛∗ (𝑡) (6)

is a second solution of (1) for 𝑡0 − 𝛿 < 𝑡 < 𝑡0 + 𝛿 satisfying (1), then for . Because

every nth-order ODE can be expressed as a system of n first- order ODE, this theorem also applies to the single nth-

order ODE.

1.4 Basics of exact first order Ordinary Differential Equations

An exact first order ODE is one of the form

Where,

An equation of the form (1) with

is said to be non exact. If

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

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

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

1.5 Basics of Special classes of Second Ordinary Differential Equations

Special classes of the second order differential equations include,

If x is missing then,

(If y is missing). A second-order linear homogeneous ODE

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

The undamped equation of simple harmonic motion is

which becomes

when damped, and

when both forced and damped.

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

The following are examples of important ordinary differential equations which commonly arise in problems of

mathematical physics.

1. 6 Bernoulli differential equation and Solution by MATHEMATICA Software

Form of the Bernoulli differential equation. i.e.

Bernoulli differential equation can solve by MATHEMATICA Software.

Programming for the solution of Bernoulli differential equation.

Programming for Bernoulli differential equation

Cell["Bernoulli", "Section"],

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Cell[BoxData["True"], "Output"]

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1.7 Bessel differential equation and Solution by MATHEMATICA Software

Form of the Bessel differential equation

Bessel differential equation can solve by MATHEMATICA Software.

Programming for the solution of Bessel differential equation.

Programming for Bessel differential equation

Cell["Bessel", "Section"],

Cell[BoxData[

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Cell[BoxData["True"], "Output"]

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1.7 Conclusion

Some special ordinary differential equations Bernoulli differential equation and Bessel differential equation can be

solve by above Programming through MATHEMATICA software.

1.8 References

[1] A.K.Awasthi,Solution of the ordinary differential equations using Mathematica software ,ToI Press; 2019.p.279 –329.

[2] A.K.Awasthi,Solution of the Anger & Baer ordinary differential equations using Mathematica software, ToI Press;

2019.p.91 –113.

Computation of Bernoulli and Bessel Differential Equation

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