Continue solutions of ODEs around the origin

psie
Messages
315
Reaction score
40
Homework Statement
Solve the following equations when ##z## is real and positive. How will the solutions change if continued one revolution around the origin in the positive direction? a) ##2zx'-x=0## and b) ##z^2x'+x=0##.
Relevant Equations
By "continued one revolution around the origin", they basically mean evaluate ##x(ze^{2\pi i})##.
What confuses me is that my solution differs from that given in the answers at the back of the book.

Solving the ODEs is fairly simple. They are both separable. After rearrangement and simplification, you arrive at ##x(z)=Cz^{1/2}## for a) and ##x(z)=De^{1/z}## for b). In both solutions, ##C## and ##D## are positive constants.

Then I'm asked to evaluate ##x(ze^{2\pi i})## and a TA claims that ##x(ze^{2\pi i})=C(ze^{2\pi i})^{1/2}=Cz^{1/2}e^{\pi i}=-Cz^{1/2}## for a) and that ##x(ze^{2\pi i})=De^{1/(ze^{2\pi i})}=De^{1/z}## for b). This is also the answer given in the back of the book. I agree for b), but for a), why isn't it true that ##x(ze^{2\pi i})=C(ze^{2\pi i})^{1/2}=C(z\cdot1)^{1/2}=Cz^{1/2}##?

Edit: This problem appears in a section on differential equations with singular points, i.e. where the matrix ##A(z)## in the corresponding system ##x'(z)=A(z)x(z)## has a singularity.
 
Last edited:
Physics news on Phys.org
Which branch of the square root are you using? In the principal branch, \arg z \in (-\pi, \pi] and \arg z^{1/2} = \frac12 \arg z \in (-\pi/2, \pi/2] so that \operatorname{Re}(z^{1/2}) \geq 0. However, in the branch in which \arg z \in (\pi, 3\pi] then \arg z^{1/2} = \frac12 \arg z \in (\pi/2, 3\pi/2] so that \operatorname{Re}(z^{1/2}) \leq 0. e^{0i} is not in this branch, but e^{2\pi i} is.

In both cases, \sqrt{re^{i\theta}} = r^{1/2}e^{i\theta/2} with r^{1/2} \geq 0.
 
Right, a branch...I forgot! I guess the book uses the latter branch, i.e. for which ##\arg z \in (\pi, 3\pi]##, since they input a real positive number and get a negative positive number. It makes more sense now. Thank you.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top