# Particular Solutions of Ordinary Differential Equations Using Discrete Symmetry Groups

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## Abstract

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## 1. Introduction

## 2. Method to Find the Discrete Point Symmetries of an ODE

## 3. Application of Discrete Symmetry Groups for Obtaining Particular Solutions of Nonlinear ODEs

#### 3.1. Procedure to Find Particular Solutions Using Discrete Symmetry Groups

#### 3.2. Particular Solutions of Some Nonlinear ODEs Using Discrete Symmetry Groups

- A third order nonlinear ODE, obtained by Whittaker [19]$${w}^{\u2034}=3{w}^{\u2033}+\frac{{w}^{\u2033}{w}^{\prime}}{w}-\frac{{w}^{{}^{\prime}2}}{w}-2{w}^{\prime},$$$$-w=f\left(x\right).$$If we take $f\left(x\right)={e}^{x}$ then it is observed that $-w={e}^{x}$ or $w=-{e}^{x}$ is the particular solution of $\left(4\right)$. The graph of particular solution $w=-{e}^{x}$ of $\left(4\right)$ is shown in Figure 1.
- Consider the Blasius equation$${w}^{\u2034}+\frac{1}{2}w{w}^{\u2033}=0,$$$${\partial}_{x},x{\partial}_{x}-w{\partial}_{w}.$$Following the method presented in [18], it is found that the discrete symmetry group of $\left(12\right)$ is $\mathsf{{\rm Y}}:(x,w)\mapsto (-x,w)$, which shows that $w=-f\left(x\right)$ i.e., $w=-x$ is the particular solution of $\left(12\right)$. The graph of particular solution $w=-x$ of $\left(12\right)$ is presented in Figure 2.
- Consider the following ODE$${w}^{\u2034}=\frac{3}{2}\frac{{w}^{{}^{\u2033}2}}{{w}^{\prime}},$$$${\partial}_{x},z{\partial}_{x},{x}^{2}{\partial}_{x},{\partial}_{w},w{\partial}_{w},{w}^{2}{\partial}_{w}.$$Using above Lie algebra and applying Hydon’s method given in [18], it is found that$${\mathsf{{\rm Y}}}_{1}:(x,w)\mapsto (\frac{1}{x},w),$$$${\mathsf{{\rm Y}}}_{2}:(x,w)\mapsto (-x,w),$$
- The discrete symmetry group of nonlinear ODE [21]$${w}^{\u2034}={w}^{\u2033}-\left({w}^{{}^{\u2033}2}\right),$$$$\mathsf{{\rm Y}}:(x,w)\mapsto (-x,\frac{1}{2}{x}^{2}-w).$$Now using $\left(10\right)$, $\mathsf{{\rm Y}}$ deduce that $\frac{1}{2}{x}^{2}-w=-x$ or $w=\frac{1}{2}{x}^{2}+x$ is a particular solution of $\left(16\right).$ The graph of particular solution $w=\frac{1}{2}{x}^{2}+x$ of $\left(16\right)$ is presented in Figure 4.
- Consider the following Chazy equation$${w}^{\u2034}=2w{w}^{\u2033}-3{w}^{{}^{\prime}2}+\lambda {(6{w}^{\prime}-{w}^{2})}^{2},$$$$(x,w)\mapsto (-\frac{1}{x},{x}^{2}w+6x),$$$${x}^{2}w+6x={c}_{1}f(-\frac{1}{x}).$$$${x}^{2}w+6x=-{c}_{1}\frac{1}{x}.$$To seek the particular solution of $\left(17\right)$, the value of ${c}_{1}$ has to be found. So the derivatives, ${w}^{\prime}$, ${w}^{\u2033}$ and ${w}^{\u2034}$ by using Equation $\left(18\right)$ are obtained and by putting values of these derivatives in Equation $\left(17\right)$, it is obtained that ${c}_{1}=0$. Thus ${x}^{2}w+6x=0$ i.e., $w=-\frac{6}{x}$ is the particular solution of $\left(17\right)$. The graph of this particular solution is shown in Figure 5.

## 4. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

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Bibi, K.
Particular Solutions of Ordinary Differential Equations Using Discrete Symmetry Groups. *Symmetry* **2020**, *12*, 180.
https://doi.org/10.3390/sym12010180

**AMA Style**

Bibi K.
Particular Solutions of Ordinary Differential Equations Using Discrete Symmetry Groups. *Symmetry*. 2020; 12(1):180.
https://doi.org/10.3390/sym12010180

**Chicago/Turabian Style**

Bibi, Khudija.
2020. "Particular Solutions of Ordinary Differential Equations Using Discrete Symmetry Groups" *Symmetry* 12, no. 1: 180.
https://doi.org/10.3390/sym12010180