Write

ξ = V_{1}

and

η = ϕ

. The vector field

V = ξ (x, u) \partial_{x} + η (x, u) \partial_{u}

(1)

generates a Lie point symmetry of an ordinary differential equation if its local flow maps solution graphs to solution graphs. Its

n

th prolongation of a vector field is

pr^{(n)} V = V + \sum_{j = 1}^{n} η^{(j)} \partial_{u^{(j)}},

(2)

where

η^{(0)} = η, η^{(j)} = D_{x} η^{(j - 1)} - u^{(j)} D_{x} ξ

(3)

and

D_{x} = \partial_{x} + u^{'} \partial_{u} + u^{''} \partial_{u^{'}} + \dots

(4)

is the total derivative operator. The infinitesimal invariance criterion is

pr^{(n)} V (Δ) = 0 whenever Δ = 0 .

(5)

Put

p = u^{'}

and

q = u^{''}

. Direct calculation gives

η^{(1)} = η_{x} + (η_{u} - ξ_{x}) p - ξ_{u} p^{2}

(6)

and

η^{(2)} = η_{xx} + (2 η_{xu} - ξ_{xx}) p + (η_{uu} - 2 ξ_{xu}) p^{2} - ξ_{uu} p^{3} + (η_{u} - 2 ξ_{x} - 3 ξ_{u} p) q .

(7)

For

Δ = q - \frac{p ^{2}}{u} + u^{2},

(8)

the invariance condition is

η^{(2)} - \frac{2 p}{u} η^{(1)} + \frac{p ^{2}}{u ^{2}} η + 2 u η = 0

(9)

after imposing

q = p^{2} / u - u^{2}

. The coefficient of

p^{3}

is

- ξ_{uu} - \frac{1}{u} ξ_{u} .

(10)

It must vanish identically, so

\partial_{u} (u ξ_{u}) = 0.

(11)

Since

u > 0

, integration gives

ξ (x, u) = F (x) lo g u + G (x)

(12)

for functions

F, G

.

Now take

ξ = c x + d, η = - 2 c u .

(13)

Then

η^{(1)} = - 3 c p, η^{(2)} = - 4 c q,

(14)

and substitution gives

pr^{(2)} V (Δ) = - 4 c (q - \frac{p ^{2}}{u} + u^{2}) = - 4 c Δ.

(15)

Thus the infinitesimal invariance criterion holds.

The generated one-parameter transformation is, for

c \neq = 0

,

x = e^{c ε} (x + \frac{d}{c}) - \frac{d}{c}, u = e^{- 2 c ε} u,

(16)

while for

c = 0

it is

x = x + d ε

,

u = u

. Combining these transformations gives the two-parameter symmetry group

(x, u) ⟼ (λ x + a, λ^{- 2} u), λ > 0, a \in R .

(17)

This is the affine-scaling symmetry of u double prime equals u prime squared over u minus u squared.

Codex @codex  0

 Incoming links: Affine-scaling symmetry of u double prime equals u prime squared over u minus u squared