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I am following the course 18.034 offered at the OCW. In the book "Ordinary Differential Equations" of Garrett Birkhoff and Gian-Carlo Rota I encountered the following problem:

For a linear fractional DE we have:

$$y''=(ad-bc)[cx^2-(a-d)xy-by^2]/(ax+by)^3$$

Discuss the domains of convexity and concavity of solutions.

According to the course solution:

Let $\alpha=a+ib, \beta=c+id $. In terms of polar coordinate functions, $$y''=\frac{Im(\alpha\beta)cos(\theta)Re((\beta+i\alpha)e^{i\theta})}{Re(\alpha e^{-i\theta})}$$

So y'' changes sign at slopes -b/a, $\infty$, $\frac{b-c}{a+d}$

I do not understand how, from the DE given, could you get to the polar coordinate function that gives the solution. Also, how do I know that y'' change signes at those slopes? I would really appreciate if you could help me

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    I am used to the term `fractional ode` as $y^{(1/2)}(x)$ where it is a fractional order of derivative. – 2017-01-12
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    letting $\alpha = a+ ib$ and $\beta = c+id$ we have $$ \alpha\beta = ac -bd + (ad+bc)i\implies \mathcal{Im}(\alpha\beta) = ad + bc $$ which is clearly not what you have. So please check your question. – 2017-01-12

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