Here is an exercise of Section 1 of Friedman Algebraic Surfaces and Holomorphic Vector Bundles.
Let $C$ be an irreducible curve whose singular locus is a single ordinary double point $x$, and let $p,q\in\tilde{C}$ be the preimage under the normalization map $\nu$ of $x$. Show that $\nu^*\omega_C=K_{\tilde{C}}\otimes\mathcal{O}_{\tilde{C}}(p+q)$.
Here is my attempt. I think it suffices to consider near $p,q$. Localize at $p$ and we have \begin{align} (\nu^*\omega_C)_p&\cong\Omega_{\mathcal{O}_{C,x}/\mathbb{C}}\otimes_{\mathcal{O}_{C,x}}\mathcal{O}_{\tilde{C},p}\\ (K_{\tilde{C}}\otimes\mathcal{O}_{\tilde{C}}(p+q))_p&\cong\Omega_{\mathcal{O}_{\tilde{C},p}/\mathbb{C}}\otimes_{\mathcal{O}_{\tilde{C},p}}\mathrm{Hom}_{\mathcal{O}_{\tilde{C},p}}(\mathfrak{m}_p,\mathcal{O}_{\tilde{C},p}). \end{align} There is then a natural map $$(\nu^*\omega_C)_p\to (K_{\tilde{C}}\otimes\mathcal{O}_{\tilde{C}}(p+q))_p. $$ Then I expect that after passing to the completion $\mathcal{O}_{\tilde{C},p}\to \widehat{\mathcal{O}}_{\tilde{C},p}$, the above map would be an isomorphism. My idea is that we need to make use of the condition $$\widehat{\mathcal{O}}_{C,x}\cong\mathbb{C}[[s,t]]/st$$ while we do not have much information of $\mathcal{O}_{C,x}$. However, I can not find theorems relating completion and differentials. The closest is $$\Omega_{\mathcal{O}_{C,x}/\mathbb{C}}\otimes_{\mathcal{O}_{C,x}}\widehat{\mathcal{O}}_{C,x}\to \Omega_{\widehat{\mathcal{O}}_{C,x}/\mathbb{C}}\to\Omega_{\widehat{\mathcal{O}}_{C,x}/{\mathcal{O}}_{C,x}}\to0$$ where I have no idea of $\Omega_{\widehat{\mathcal{O}}_{C,x}/{\mathcal{O}}_{C,x}}$. Should I expect $\Omega_{\widehat{\mathcal{O}}_{C,x}/{\mathcal{O}}_{C,x}}=0$?
The exercise has another two questions. I think they are attainable if the above is resolved. I am happy to hear any tricks anyway.
Which local sections of this line bundle are the pullbacks of sections of $\omega_C$? Analyze a cusp singularity similarly.