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© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
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.
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 406
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
ODEs.
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
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 407
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
(7)
Where,
(8)
An equation of the form (1) with
(9)
is said to be non exact. If
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 408
(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
1.5 Basics of Special classes of Second Ordinary Differential Equations
Special classes of the second order differential equations include,
(13)
If x is missing then,
(14)
(If y is missing). A second-order linear homogeneous ODE
(15)
for which can be transformed to one with constant coefficients.i.e.
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 409
(16)
The undamped equation of simple harmonic motion is
(17)
which becomes
(18)
when damped, and
(19)
when both forced and damped.
Form of the Systems with constant coefficient. i.e.
(20)
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.
(21)
Bernoulli differential equation can solve by MATHEMATICA Software.
Programming for the solution of Bernoulli differential equation.
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 410
Programming for Bernoulli differential equation
Cell["Bernoulli", "Section"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"(",
RowBox[{"soln", "=",
RowBox[{"DSolve", "[",
RowBox[{
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RowBox[{"y", "'"}], "[", "x", "]"}], "+",
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RowBox[{"y", "[", "x", "]"}]}]}], "\[Equal]",
RowBox[{
RowBox[{"q", "[", "x", "]"}],
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RowBox[{"y", "[", "x", "]"}], "^", "n"}]}]}]}], ",",
RowBox[{"y", "[", "x", "]"}], ",", "x"}], "]"}]}], ")"}], "//",
"Timing"}]], "Input"],
Cell[BoxData[
RowBox[{
RowBox[{"Solve", "::", "\<\"ifun\"\>"}], ":",
" ", "\<\"Inverse functions are being used by \\!\\(Solve\\), so some \
solutions may not be found; use Reduce for complete solution information. \\!\
\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"Solve::ifun\\\"]\\)\"\>"}]], "Message"],
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 411
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"0.23225600000000668`", " ", "Second"}], ",",
RowBox[{"{",
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RowBox[{
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RowBox[{
RowBox[{"-",
RowBox[{"p", "[", "K$63157", "]"}]}],
RowBox[{"\[DifferentialD]", "K$63157"}]}]}]}]], " ",
RowBox[{
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 412
SubsuperscriptBox["\[Integral]", "1", "x"],
RowBox[{
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{
RowBox[{"-",
RowBox[{"(",
RowBox[{"1", "-", "n"}], ")"}]}], " ",
RowBox[{
SubsuperscriptBox["\[Integral]", "1", "K$63173"],
RowBox[{
RowBox[{"-",
RowBox[{"p", "[", "K$63157", "]"}]}],
RowBox[{"\[DifferentialD]", "K$63157"}]}]}]}]], " ",
RowBox[{"q", "[", "K$63173", "]"}]}],
RowBox[{"\[DifferentialD]", "K$63173"}]}]}]}], "-",
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SubsuperscriptBox["\[Integral]", "1", "x"],
RowBox[{
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SuperscriptBox["\[ExponentialE]",
RowBox[{
RowBox[{"-",
RowBox[{"(",
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 413
RowBox[{"1", "-", "n"}], ")"}]}], " ",
RowBox[{
SubsuperscriptBox["\[Integral]", "1", "K$63173"],
RowBox[{
RowBox[{"-",
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RowBox[{"\[DifferentialD]", "K$63173"}]}]}]}]}], ")"}],
FractionBox["1",
RowBox[{"1", "-", "n"}]]]}], "}"}], "}"}]}], "}"}]], "Output"]
}, Open ]],
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"//", "FullSimplify"}]], "Input"],
Cell[BoxData["True"], "Output"]
}, Open ]]
}, Open ]]
1.7 Bessel differential equation and Solution by MATHEMATICA Software
Form of the Bessel differential equation
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 414
(22)
(23)
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[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"(",
RowBox[{"soln", "=",
RowBox[{"DSolve", "[",
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RowBox[{"ode", "=",
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RowBox[{
RowBox[{"x", "^", "2"}], " ",
RowBox[{
RowBox[{"y", "''"}], "[", "x", "]"}]}], "+",
RowBox[{"x", " ",
RowBox[{
RowBox[{"y", "'"}], "[", "x", "]"}]}], "+",
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 415
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"\[Lambda]", "^", "2"}], " ",
RowBox[{"x", "^", "2"}]}], "-",
RowBox[{"n", "^", "2"}]}], ")"}],
RowBox[{"y", "[", "x", "]"}]}]}], "\[Equal]", "0"}]}], ",",
RowBox[{"y", "[", "x", "]"}], ",", "x"}], "]"}]}], ")"}], "//",
"Timing"}]], "Input"],
Cell[BoxData[
RowBox[{"{",
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RowBox[{
RowBox[{
RowBox[{"BesselJ", "[",
RowBox[{"n", ",",
RowBox[{"x", " ", "\[Lambda]"}]}], "]"}], " ",
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 416
RowBox[{"C", "[", "1", "]"}]}], "+",
RowBox[{
RowBox[{"BesselY", "[",
RowBox[{"n", ",",
RowBox[{"x", " ", "\[Lambda]"}]}], "]"}], " ",
RowBox[{"C", "[", "2", "]"}]}]}]}], "}"}], "}"}]}], "}"}]], "Output"]
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Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"ode", "/.",
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RowBox[{"Function", "[",
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RowBox[{"[",
RowBox[{"1", ",", "1", ",", "2"}], "]"}], "]"}], "]"}]}], "]"}]}]}],
"//", "FullSimplify"}]], "Input"],
Cell[BoxData["True"], "Output"]
}, Open ]]
}, Open ]]
1.7 Conclusion
© 2018 IJRAR December 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)
IJRAR1BIP073 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 417
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.