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I am currently reading a book says that the ODE $$f'' + \frac{1}{x}f' +cf =0$$ has unique solution by Picard-Lindelöf or Cauchy-Lipschitz theorem given $f(0)=1, f'(0) =0.$

I tried to rewrite this as a first order ODE by defining: $$g = \begin{bmatrix} f'\\ f \end{bmatrix} \quad A= \begin{bmatrix} \frac{1}{x} &c\\ 1 &0 \end{bmatrix}$$ Then I have $$g' = Ag$$

But I found that $A$ is not bounded, so how can one apply the uniqueness theorem to this ODE? Many thanks!

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It's not bounded, but it is continuous and locally Lipschitz with respect to $g$ in the complement of $\{0\}$, and that's all you need for local existence and uniqueness. Of course the equation isn't defined at $x=0$, so you don't expect anything in a domain that includes $0$.

Actually, since the equation is linear, you have not just local but global existence and uniqueness in $(0,\infty)$ or $(-\infty,0)$.

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