Proper way to solve first order circuit with voltage dependent voltage source
My solution: Consider \$v_L\$ as the voltage on inductor (in accordance with the direction of \$i\$) and \$v_A\$ as the voltage on upper node (called here node \$A\$). Applying the KCL on this node:

$$ \frac{v_A-v_L}{4}+ \frac{v_A-2v_x}{1} + \frac{v_A}{4}=0 $$

If
$$ \left\{\begin{matrix}v_L=2 \frac{\mathrm{d}i }{\mathrm{d} t} \\v_x=-4i \end{matrix}\right. $$

Then

$$ \frac{\mathrm{d}i }{\mathrm{d} t}-16i-3v_a=0 $$

Replacing \$ v_a=2\frac{\mathrm{d}i }{\mathrm{d} t}+4i \$ leads to:

$$ 5\frac{\mathrm{d}i }{\mathrm{d} t}+28i=0 $$

So, the current \$i(t)\$ is given by (in Ampere):
$$i(t) = 5e^{-5.6t} A$$

Remebering that \$ v_x=-4i \$, this voltage is given by (in Volt):

$$v_x(t) =-20e^{-5.6t} $$