# Initial Value Problem, possible error in question

I have the following problem (1.4 from Qingkai Kong’s Short Course in ODEs) in my differential equations homework, but I think that the initial condition given makes it unsolvable. Please inspect my work and tell me if I am mistaken somewhere.

"Show that for $\alpha \in (0,1]$ the IVP \begin{equation*} x \,' \, = \, \begin{cases} x\, \ln^{\,\alpha}|x|, & x \neq 0,\\ 0, & x = 0; \end{cases} \,\,\,\,\, x(0) = 0 \end{equation*} has a unique solution,’'

I am taking cases on whether $\alpha < 1$ or $\alpha = 1$. In either case I separate variables. To integrate $\frac{dx}{x \ln^{\alpha}(x)}$, I make the substitution $u = \ln |x|$.

First, take $\alpha \in (0,1)$. Then there is a constant $c$ so that $$t + c = \int \frac{du}{u^{\alpha}} = \frac{1}{1-\alpha}u^{1-\alpha} = \frac{1}{1-a}(\ln|x|)^{1-\alpha}.$$ I believe that it is impossible to satisfy the given initial condition, because substituting $0$ for each of $x$ and $t$, this would imply that $c$ was $- \infty$.

In the $\alpha = 1$ case, there is some constant $c$ so that $$t + c = \int \frac{du}{u} = \ln|\ln |x||.$$ Note that if I take $x(0) = 0$, then I cannot even take a natural log twice on the right side.

I suspect that there was an error in the statement of the problem (which is not implausible, as the textbook was published in 2014). Perhaps it ought to have given $x(0) = 1$ as an initial condition instead? This would make sense in the case that $\alpha < 1$ case, as it would give a constant of integration equal to $0$ (which I find is often the case in such problems). Then in the $\alpha = 1$ case, I could say that $x \equiv 0$ is a solution.

• What is the differential equation at $x=0$ and what can be said about that? Also what would you get if you write $x$ as a function of $t$ while $c=-\infty$? – Kwin van der Veen Sep 7 '15 at 11:13
• When $x=0$, $x’ = 0$ as well. I suppose I could take $x \equiv 0$ as a solution, but surely this can’t be what the author intended?! – Jordan Green Sep 7 '15 at 13:53