Again differentiating w.r.t. $x$, we get
$$
y^{\prime \prime}=y^{\prime}+2 B e^{2 x}+6 C e^{3 x}
$$
From Eq. (ii)
$$
\begin{array}{rlrl}
B e^{2 x} & =y^{\prime}-y-2 C e^{3 x} \\
\therefore \quad & y^{\prime \prime} =y^{\prime}+2 y^{\prime}-2 y-4 C e^{3 x}+6 C e^{3 x}
\end{array}
$$

Again differentiating w.r.t. $x$, we get
$$
y^{\prime \prime \prime}=3 y^{\prime \prime}-2 y^{\prime}+6 C e^{3 x}
$$
From Eq. (iii)
$$
\begin{aligned}
& 2 C e^{3 x}=y^{\prime \prime}-3 y^{\prime}+2 y \\
& \therefore \quad y^{\prime \prime \prime}=3 y^{\prime \prime}-2 y^{\prime}+3\left(y^{\prime \prime}-3 y^{\prime}+2 y\right) \\
& \Rightarrow \quad y^{\prime \prime \prime}=6 y^{\prime \prime}-11 y^{\prime}+6 y \\
& \Rightarrow \quad y^{\prime \prime \prime}-6 y^{\prime \prime}+11 y^{\prime}-6 y=0 \\
&
\end{aligned}
$$