Step 1: Classify the two amines. $C_2H_5NH_2$ is a primary amine (one $N-H$ group attached to alkyl); $(C_2H_5)_2NH$ is a secondary amine (two alkyl groups on N). Step 2: Carbylamine test distinguishes them. Only primary amines react with $CHCl_3$ and alc. $KOH$ to produce a foul-smelling isocyanide: $RNH_2 + CHCl_3 + 3KOH \rightarrow RNC + 3KCl + 3H_2O$. Secondary and tertiary amines give no reaction. Step 3: Apply test. $C_2H_5NH_2$ gives a distinct foul smell (positive test); $(C_2H_5)_2NH$ gives no response. This cleanly distinguishes the two. \[ \boxed{CHCl_3 + KOH \text{ (ethanolic)}} \]