Direct Energy, 2018a

328 14.4 Derivation of the Innitesimal Generators
There are three possible independent solutions for ξ. They are ξ =1, ξ = t,
and ξ = y. So, we found three innitesimal generators.
U 1 = ∂ t (14.86)
U 2 = t∂ t (14.87)
U 3 = y∂ t (14.88)
Case 2 with ξ =0: Suppose ξ =0. What solutions can be found for η?
Equation 14.79 to Eq. 14.82 simplify.
∂ tt η =0 (14.89)
∂ yt η =0 (14.90)
∂ yy η =0 (14.91)
There are three possible independent solutions for η. They are η =1, η = y,
and η = t. So, we found three more innitesimal generators.
U 4 = ∂ y (14.92)
U 5 = y∂ y (14.93)
U 6 = t∂ y (14.94)
Case 3 where both ξ and η are nonzero: From Eq. 14.79, we can write
Here, b is a function of only y, not t. Therefore,
which is not a function of t.
From Eq. 14.82, we can write
Here, g is a function of only t, not y. Therefore,
which is not a function of y. Nowuse Eq. 14.80.
η =(c 1 + c 2 t) b(y). (14.95)
∂ yt η = c 2 ∂ y b(y) (14.96)
ξ =(c 3 + c 4 y) g(t). (14.97)
∂ yt ξ = c 4 ∂ t g(t) (14.98)
2c 2 ∂ y b(y) − (c 3 + c 4 y) ∂ tt g =0 (14.99)