The Frobenius method on a second-order homogeneous linear ODEs

Haarsa, P.; Pothat, S.

doi:10.12988/astp.2014.4798

Cited by 3 publications

(3 citation statements)

References 2 publications

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“…This well-known differential equation was first studied in connection with heavy chain oscillations and circular membrane vibrations [3]. In addition, this equation is seen in heat transfer, stress analysis, fluid mechanics, and vibrations [6]. Linear ODEs with constant coefficients could simply be solved with functions recognized from [3].…”

Section: Introductionmentioning

confidence: 99%

“…Laplace's transform approach, known as mathematician Pierre-Simon Laplace, is a suitable integral transform approach to find solutions set for the BDE for certain initial conditions [1]. Another suitable approach is the power series approach, which is a very common approach for solving linear ODEs [6]. The BDE is frequently solved on the Frobenius approach based on the power series [7].…”

Section: Introductionmentioning

confidence: 99%

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Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

Azizi,

Hayat,

Rohani

et al. 2024

J. Res. Appl. Sci. Biotechnol.

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The BDE (Bessel differential equation) is a second-order linear ordinary differential equation (ODE), and it is considered one of the most significant differential equations because of its extensive applications. The solutions of this differential equation are called Bessel functions. These solutions can be expressed in terms of hypergeometric functions, and in this study, the Bessel functions of the first kind are worked. Hypergeometric functions are the sum of a hypergeometric series. A series ∑▒u_n is known as hypergeometric when the ratio u_(n+1)⁄u_n is a rational function of n. BDE has coefficients with variables; therefore, it is solved by the Frobenius approach. We implement some mathematical steps based on the definition of hypergeometric functions to express the solutions in terms of hypergeometric function. The result shows that the solution of the BDE in terms of hypergeometric function is f(x)=a_0 x^p (_1^ )F_2 (1;1+(p+ρ)/2,1+(p-ρ)/2;x^2⁄4) and the two linearly independent solutions for the BDE of order ρ in terms of hypergeometric function are 〖f_1 (x)=a〗_0 x^ρ (_0^ )F_1 (1+ρ;x^2⁄4) and〖〖 f〗_2 (x)=a〗_0 x^ρ (_0^ )F_1 (1-ρ;x^2⁄4).

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

Azizi,

Hayat,

Rohani

et al. 2024

J. Res. Appl. Sci. Biotechnol.

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“…The Frobenius method enables us to make use of powers series solution to solve a different equation, given that ( ) are themselves analytic at 0 or being analytic elsewhere. In this case both limit exist at 0 and are finite[20]. Definition 2.2.…”

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confidence: 99%

Maximum Interval of Stability and Convergence of Solution of a Forced Mathieu’s Equation

Eze¹,

Obasi²,

Ujumadu³

et al. 2020

WJM

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This paper investigates the maximum interval of stability and convergence of solution of a forced Mathieu's equation, using a combination of Frobenius method and Eigenvalue approach. The results indicated that the equilibrium point was found to be unstable and maximum bounds were found on the derivative of the restoring force showing sharp condition for the existence of periodic solution. Furthermore, the solution to Mathieu's equation converges which extends and improves some results in literature.

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Frobenius Method for Solving Second-Order Ordinary Differential Equations

Torabi¹

2020

JAMP

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As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point 0 t = and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.

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The Frobenius method on a second-order homogeneous linear ODEs

Abstract: In this paper, we show that the one solution of the second-order homogeneous linear differential equation can be obtained by using the method of Frobenius.Mathematics Subject Classification: 34B05, 34G10, 34A30, 35L05

Cited by 3 publications

References 2 publications

Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

Maximum Interval of Stability and Convergence of Solution of a Forced Mathieu’s Equation

Frobenius Method for Solving Second-Order Ordinary Differential Equations

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