A Linear Approximation of the Non-linear Modified Langumir and Van der Pol Differential Equations by the Application of the Generalized Sundman Transformation
DOI:
https://doi.org/10.56919/usci.2434.038Keywords:
Linearization, Modified Langumir, Differential Equation, Van der Pol, Generalized Sundman TransformationAbstract
Study’s Excerpt:
- The modified Langumir and Van der Pol nonlinear ordinary differential equations are solved by the GST.
- These equations are very significant and useful in many areas of human life.
- The GST method linearizes the equations into forms that can be solved, resulting in 3u''+4u'+2=0 and u''+u'+2=0, respectively.
- Here, the GST technique yielded innovative and workable analytical solutions.
Full Abstract:
The non-linear ordinary differential equations of Langumir and Van der Pol are challenging to solve analytically. Thus, this work aims to convert these non-linear equations into linear form so that they may be easily solved. Assuming that the coefficients of the two equations meet the linearizability requirements, they are presented in the appropriate linearizable formats. After achieving this, the generalized Sundman transformation was used to linearize the equations. The formulae defines the nonpoint transformation known as the generalized Sundman transformation (GST). Basic solutions for the two equations were obtained upon application of the GST. The conventional approach of variation of parameters was used to solve the linear equations that emerged from the linearization process.
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