How can this circuit be solved with the node method?
I think a problem with your work is that you've defined \$I_1\$ as a tautology. Of course \$2\cdot I_1\$ can be defined the way you wrote it. But it doesn't add any information to the problem. It simply re-states things.

Here's a way to write it:

$$\begin{align*}
I_1&amp;=\frac{V_1-V_2}{R_1}\\\\
\frac{V_2}{R_1}+\frac{V_2}{R_2}+\frac{V_2}{R_3}&amp;=\frac{V_1}{R_1}+\frac{V_3}{R_2}+\frac{V_4}{R_3}\\\\
\frac{V_3}{R_2}+\frac{V_3}{R_4}+\frac{V_3}{R_5}&amp;=\frac{V_2}{R_2}+\frac{V_4}{R_4}+\frac{V_5}{R_5}\\\\
\frac{V_4}{R_3}+\frac{V_4}{R_4}+2\cdot I_1&amp;=\frac{V_2}{R_3}+\frac{V_3}{R_4}\\\\
\frac{V_5}{R_5}+\frac{V_5}{R_6}&amp;=2\cdot I_1 + \frac{V_3}{R_5}+\frac{0\:\text{V}}{R_6}
\end{align*}$$

You should be able to solve for the five unknowns, \$V_2\$, \$V_3\$, \$V_4\$, \$V_5\$, and \$I_1\$ given the knowns (which includes \$V_1\$.)