I am following Olver's book Applications of Lie Groups to Differential Equations to solve the second order ODE: $\hspace{5mm}y''=0 \hspace{5mm} \text{where} \hspace{5mm} y = f(x)$
I have found the generating vectors $X_1 \hspace{3mm} \text{to} X_8$ ( Olver uses v to denote the vectors rather than X ).
In Olver's text, once the commutator table is found, then, through exponentiation, the groups are found.
I need help with exponentiation.
Here is what I have:
From Olver
- If $\mathbf{v} = \partial x \text{ then } \exp(\epsilon \mathbf{v})x = \exp(\epsilon \partial x )x = x + \epsilon $
- If $ \mathbf{v} = x \partial x \text{ then } \exp(\epsilon x \partial x) = e^\epsilon x $
- If $ \mathbf{v} = a \partial x \text{ where } a \text{ is any constant, then } \exp(\epsilon \mathbf{v})x = x + \epsilon a$
My vectors are as follows:
$$ \begin{eqnarray} X_1=xy\partial x+y^2\partial y \hspace{10mm} & X_2=y \partial x \hspace{10mm} &X_3=x^2 \partial x +xy \partial y \hspace{10mm} &X_4=y \partial y \\ \\ X_5=x \partial y \hspace{10mm} &X_6=\partial y \hspace{10mm} &X_7=x \partial x \hspace{10mm}& X_8=\partial x\\ \end{eqnarray} $$
I am able to find the following groups through exponentiation (based on what I am reading in Olver): $$ \begin{eqnarray} G_8: & f(x + \epsilon) \\ \\ G_7: & f(e^\epsilon x) \\ \\ G_6: & f(x) - \epsilon \\ \\ G_4: & \frac{f(x)}{e^\epsilon} \end{eqnarray} $$
Note $ G_a = \text{ the group found by exponentiation of } X_a $
I need to know if the work I have is correct. I also need to know how to find the other groups.
Please help.