1. Introduction to Symmetry Analysis
Chapter 8 - Ordinary Differential Equations
Brian Cantwell
Department of Aeronautics and Astronautics
Stanford University

2. 8.1 Extension of Lie Groups in the Plane

3. The Extended Transformation is a Group
Two transformations of the extended group
Compose the two transformations

4. The last relation is rearranged to read
Differentiating F and G gives
Comparing the expressions in parentheses we have
The composed transformation is in exactly the same form as the original transformation!

5. Finite transformation of the second derivative
The twice extended finite transformation is

6. Finite transformation of higher derivatives

7. The p-th order extended group is
where

8. Infinitesimal transformation of the first derivative
Recall the infinitesimal transformation of coordinates
where
Substitute
Expand and retain only the lowest order terms

9. The once-extended infinitesimal transformation in the plane is
where the infinitesimal function fully written out is

10. Infinitesimal transformation of the second derivative
Expand and retain only the lowest order terms
where

11. The infinitesimal function transforming third derivatives is

12. The infinitesimal transformation of higher order derivatives
Expand and retain only the lowest order terms. The p times extended infinitesimal transformation is
where

13. 8.2 Expansion of an ODE in a Lie Series - the Invariance Condition for ODEs

15. The characteristic equations associated with extended groups are

16. Construction of the general first order ODE that admits a given group - the Ricatti Equation
Let the first integral of the group be
In principle we can solve for either x or y. Assume we solve for y. The equation takes the following form.
where

17. The general solution of the Ricatti equation can always be determined if a particular solution of the equation can be found. A particular solution in this case is
To demonstrate take the differential of f.
Dividing by yields the Ricatti equation in terms of f.
Now let.
and work out the equation that governs h[x].

19. The general second-order ordinary differential equation
is invariant under the twice-extended group if and only if

20. Consider the case of the simplest second-order ODE
The invariance condition is
Fully written out the invariance condition is
For invariance this equation must be satisfied subject to the condition that y is a solution of

21. The determining equations of the group are
These equations can be used to work out the unknown infinitesimals.

22. Assume that the infinitesimals can be written as a multivariate power series
Insert these series into the determining equations

23. The coefficients must satisfy the following algebraic system
Finally the infinitesimals are

29. The software package used on Yxx = 0

36. Example 8.2

39. Example 8.2 – Using the software

49. The two parameter group of the Blasius equation
The invariance condition

50. Written out the invariance condition is

51. Now gather coefficients of like products of derivatives of y

52. The function y[x] is a solution of the Blasius equation
The function y[x] is a solution of the Blasius equation. This is a constraint on the invariance condition that can be used to eliminate the third derivative.

53. Further simplify
and

54. Finally the determining equations are
From which the two parameter group of the Blasius equation is determined to be