How to find the time constant of a RC circuit with dependent sources
Well, we have the following circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

Using KCL, we can write:

$$\text{I}_1=\text{I}_2+\text{I}_3\tag1$$

Using KVL, we can write:

$$
\begin{cases}
\text{I}_1=\frac{\text{V}_3}{\text{R}_2}\\
\\
\text{I}_2=\frac{\text{V}_2-\text{V}_3}{\text{R}_1}\\
\\
\text{I}_3=\frac{\text{V}_x-\text{V}_3}{\text{R}}\\
\\
\text{V}_1=\text{V}_2-\text{V}_x
\end{cases}\tag2
$$

Now, using the fact that \$\text{R}\$ is not a resistor but a capacitor and assuming that the initial condition is equal to zero. We can set:

$$\text{R}\space\space\space\to\space\space\space\frac{1}{\text{sC}}\tag3$$

Besides that we also know that:

$$\text{V}_1=-\alpha\text{V}_3\tag4$$