By C.-G. Schmidt
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I) Show that 33 is invariant under (ii) Show that under the change of variables the original equation becomes 7. (i) Show that is invariant under (ii) Show that under the change of variables the original equation becomes 8. 41) . 35 Chapter 2 Ordinary Differential Equations In this chapter, we focus on the symmetries of ordinary differential equations. 28), which were all transformed to equations that were separable and independent of . 1). 28) Equation Transformation Lie Group We now ask, where did these transformations come from and how do they relate to Lie groups?
52 , , where is an arbitrary 53 constant. 47. 49. This is often the case. We usually try the following forms (see the exercises): Exercises 1. For the given Lie group, find the corresponding infinitesimals and 2. For the given infinitesimals and the corresponding Lie group. 54 . , find 3. Find infinitesimal transformations leaving the following ODEs invariant. Use these to find a change of variables and reduce the original ODE to one that is separable and solve the equation. Hints: Try: a. b. c.
The answer is yes—they are all invariant under some infinitesimal transformation. 56 Thus, where Choosing Calculating and and are arbitrary functions. 53 and simplifying gives which is a separable equation. 57 becomes 58 which is separable. 2 Bernoulli Equation Equations of the form are called Bernoulli equations. 58. 59 We best show this via an example. 60 then becomes which is separable. 62 gives giving the infinitesimals and . 63 with and arbitrary functions of their arguments. 61 , we obtain or a separable ODE.
Arithmetik Abelscher Varietaeten mit komplexer Multiplikation by C.-G. Schmidt