Peano on Wronskians: A Translation - Appendix 1: Student Tasks

There are many excellent opportunities for additional questions or projects for students from these translations. Here are just a few that we think work particularly well. They are roughly ordered from simple exercises to more advanced projects.

1)   Consider the functions \( x \) and \( y\) described in Sur le déterminant Wronskien.

(a) Plot the functions \( x \) and \( y\) on \( (x,t) \) and \( (y,t) \) axes and give a geometric argument why the two functions are indeed linearly independent.

(b) Parametrically plot \( x \) and \( y, \) and explain why this means that \( x \) and \( y\)  are linearly independent.

(c) Calculate the Wronskian for these two functions, and show that you always get a determinant of \( 0. \) It may be easier to consider separately the cases when \( t \) is negative, positive, and \( 0. \)

2)   Consider the functions \( x \) and \( y\) described in Sur les Wronskiens.

(a) Plot the functions \( x \) and \( y\) on \( (x,t) \) and \( (y,t) \) axes and give a geometric argument why the two functions are indeed linearly independent.

(b) Parametrically plot \( x \) and \( y, \) and explain why this means that \( x \) and \( y\) are linearly independent. Show that you indeed always get a zero Wronskian.

3)    Prove Peano’s Second Theorem in the case when \( n=2. \)

4)    Building from Peano’s examples, construct an example of three functions that are not linearly dependent, but have a zero Wronskian.

5)    In [B1, p. 120], Bocher gave the following example of two functions that are linearly independent yet have a zero Wronskian. Prove this is true.

Image used with permission from the American Mathematical Society.

6)    In [B3, p. 143], Bocher gave the following example of three functions that are linearly independent yet have a zero Wronskian. Prove this is true.

Image used with permission from the American Mathematical Society.

7)    In [C, p. 292], Curtiss gave the following example of four functions that have a zero Wronskian on any interval containing the origin and are linearly dependent, but for which none of the Wronskians of three functions vanish simultaneously at a point. Prove this is true.

Image used with permission of Göttingen State and University Library.

8)    Show that the elementary function \[ \phi (t) = [1 - \psi (t)]\,{\frac{t+\psi(t)}{{\rm mod.}t+\psi(t)}},\] given by Mansion in his first footnote (see page 2; \( \psi \) is defined there), does indeed have the properties of the function \( \phi(t) \) described by Peano in Passage 2.

9)    Show that the function \[ \phi (t) = \frac{2}{\pi} {\int_0^\infty} \frac{\sin tx}{x}\,dx \] has the properties of the original function \( \phi(t) \) described by Peano in Passage 2.

10)  Provide the translation of Mansion’s proof to students.

(a) Ask students to reconstruct the argument and find the problems in Mansion's proof.

(b) In particular, assuming that the induction hypothesis is correct, correct the alternating sign typos in Mansion’s proof.

11)  Provide the translation of Jordan’s proof to students and ask them to reconstruct the argument and find the problems in the proof.

12)  Read the papers of Bocher and others and write a report on the various hypotheses that fix the proposition that zero Wronskian implies linear dependence.