Nodal analysis in circuit contain voltage controlled voltage source

I have a circuit below and I need to determine the potential at $$\v_1\$$ and $$\v_2\$$. I have 3 questions:

simulate this circuit – Schematic created using CircuitLab

1. How do I use nodal analysis at $$\v_2\$$, or how do I solve this without nodal analysis?
2. I have tried to do this:

At node $$\v_1\$$: $$\-1+\frac{v_1-v_2}{10}+\frac{v_1}{5}+\frac{v_1-v_2}{15}=0\$$

At node $$\v_2\$$: $$\v_2 = 2v_x = 2\frac{v_2-v_1}{10} = \frac{v_2-v_1}{5}\$$

$$=> \frac{11}{30}v_1-\frac{1}{6}v_2=1; \frac{1}{5}v_1+\frac{4}{5}v_2=0$$

$$=> v_1=\frac{120}{49}V; v_2=-\frac{30}{49}V$$

This is wrong, but when I ran a simulator and I accidentally set the function in VCVS is $$\0.2(v_a-v_b)\$$, then I have the same result. Why does this happen?

1. (This question is unrelate to 2 questions above, but just for curious) When I tried to substitute the $$\I_2\$$ source with a voltage source in a simulator, the simulation stop immediately, but this does not happen with $$\I_1\$$ source. Why does this happen?

Any help would be appreciated.

• Asking about simulators and nodal analysis in one question is a step too far. Mar 25 at 9:35
• Is a and b and v1 all connected together, as it looks they are to me?
– jonk
Mar 25 at 9:38
• @jonk a and b are on 2 side of the 10 ohm resistor. Mar 25 at 9:40
• I see a lot of dotted connections in your schematic. Way too many, to my eye.
– jonk
Mar 25 at 9:41
• @jonk sorry, this is my first time using circuit lab, let me edit the schematic. Mar 25 at 9:42

You said that $$\v_x = \frac{v_2 - v_1}{10}\$$