# New formula for solution of one class of linear differential equations of the second order with the variable coefficients

• Avyt Asanov Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan
• Kanykei Asanova Institute of Mathematics, National Academy of Sciences of Kyrgyz Republic, Bishkek, Kyrgyzstan
Keywords: The linear differential equation, the second order, the variable coefficients,the new formula for the common solution, Cauchy problem, examples.

### Abstract

Exact solutions for linear and nonlinear differential equations play an important role
in theoretical and practical research. In particular many works have been devoted to
finding a formula for solving second order linear differential equations with variable
coefficients. In this paper we obtained the formula for the common solution of the
linear differential equation of the second order with the variable coefficients in the
more common case. We also obtained the new formula for the solution of the Cauchy
problem for the linear differential equation of the second order with the variable
coefficients.Examples illustrating the application of the obtained formula for solving
second-order linear differential equations are given.
Key words: The linear differential equation, the second order, the variable
coefficients,the new formula for the common solution, Cauchy problem, examples.

### References

(1) Polyanin A. D. and Zaitsev V.F. (2003) Handbook of Exact Solutions for Ordinary Differential Equations.} Second Edition. Chapman and Hall/CRC A CRC Press Company, Boca Raron London New York Washington, D.C.

(2) Tada T. and Saiton S. (2004) A method by separation of variables for the first order nonlinear ordinary differential equations}, - J. of Analysis and Applications. 2, pp.51-63.

(3) Tada T. and Saiton S.( 2005) A method by separation of variables for the second order ordinary differential equations}, International J. of Mathematical Sciences. Jan. Volume 3. No.2, pp.289-296.

(4) Walter W. (1998) Ordinary Differential Equations}, Graduate Texts in Mathematics. Springer.

(5) Avyt Asanov, M.Haluk Chelik and Ruhidin Asanov (2011) Formulas for Solution of the Linear Differential Equations of the Second Order with the Variable Coefficients},-Journal of Mathematics Research, 3(3), pp.32-39.

(6) Avyt Asasnov,M.Haluk Chelik and Ruhidin Asanov (2012) One Formula for Solution of the Linear Differential Equations of the Second order with the Variable Coefficients}, -Global Jounnal of Pure and Applied Mathematics, 8(3)), pp.321-328.
Published
2020-06-30
How to Cite
Asanov, A., & Asanova, K. (2020). New formula for solution of one class of linear differential equations of the second order with the variable coefficients . Transactions on Machine Learning and Artificial Intelligence, 8(3), 61-68. https://doi.org/10.14738/tmlai.83.8402
Section
Articles