1.2.22.1 Terminology used and high level introduction
1.2.22.2 Introduction
1.2.22.3 Outline of the steps in solving a differential equation using Lie symmetry method
1.2.22.4 Finding \(\xi \left ( x,y\right ) ,\eta \left ( x,y\right ) \) knowing the first order ode type. Table lookup method.
1.2.22.5 Finding \(\xi \left ( x,y\right ) ,\eta \left ( x,y\right ) \) from linearized symmetry condition
1.2.22.6 Moving to canonical coordinates \(R,S\)
1.2.22.7 Definitions and various notes
1.2.22.8 Closer look at orbits and tangent vectors
1.2.22.9 Selection of ansatz to try
1.2.22.10 Examples
1.2.22.11 Example 1 on how to find Lie group \(\left ( \cc@accent {"7016}{x},\cc@accent {"7016}{y}\right ) \) given Lie infinitesimal \(\xi ,\eta \)
1.2.22.12 Example how to find Lie group \(\left ( \cc@accent {"7016}{x},\cc@accent {"7016}{y}\right ) \) given canonical coordinates \(R,S\)
1.2.22.13 Example \(y^{\prime }=\frac {y}{x}+x\)
1.2.22.14 Example \(y^{\prime }=xy^{2}-\frac {2y}{x}-\frac {1}{x^{3}}\)
1.2.22.15 Example \(y^{\prime }=\frac {y+1}{x}+\frac {y^{2}}{x^{3}}\)
1.2.22.16 Example \(y^{\prime }=\frac {y-4xy^{2}-16x^{3}}{y^{3}+4x^{2}y+x}\)
1.2.22.17 Example \(y^{\prime }=\frac {-y^{2}}{e^{x}-y}\)
1.2.22.18 Example \(y^{\prime }=\frac {x\sqrt {1+y}+\sqrt {1+y}+1+y}{1+x}\)
1.2.22.19 Example \(y^{\prime }=\frac {-y}{2x-ye^{y}}\)
1.2.22.20 Example \(y^{\prime }=\frac {-1-2yx}{x^{2}+2y}\)
1.2.22.21 Example \(y^{\prime }=3\sqrt {yx}\)
1.2.22.22 Example \(y^{\prime }=4\left ( yx\right ) ^{\frac {1}{3}}\)
1.2.22.23 Example \(y^{\prime }=2y+3e^{2x}\)
1.2.22.24 Example \(y^{\prime }=\frac {1}{3}\frac {2y+y^{3}-x^{2}}{x}\)
1.2.22.25 Example \(y^{\prime }=3-2\frac {y}{x}\)
1.2.22.26 Example \(y^{\prime }=\frac {-3+\frac {y}{x}}{-1-\frac {y}{x}}\)
1.2.22.27 Example \(y^{\prime }=\frac {1+3\left ( \frac {y}{x}\right ) ^{2}}{2\frac {y}{x}}\)
1.2.22.28 Example \(y^{\prime }=\frac {y}{x}+\frac {1}{x}F\left ( \frac {y}{x}\right ) \)
1.2.22.29 Example \(y^{\prime }=\frac {y}{x}+\frac {1}{x}e^{-\frac {y}{x}}\)
1.2.22.30 Example \(y^{\prime }=\frac {1-y^{2}+x^{2}}{1+y^{2}-x^{2}}\)
1.2.22.31 Example \(y^{\prime }=-\frac {1}{4}xe^{-2y}+\frac {1}{4}\sqrt {\left ( e^{-2y}\right ) ^{2}x^{2}+4e^{-2y}}\)
1.2.22.32 Example \(y^{\prime }=\frac {y-xf\left ( x^{2}+ay^{2}\right ) }{x+ayf\left ( x^{2}+ay^{2}\right ) }\)
1.2.22.33 Alternative form for the similarity condition PDE