Three coupled differential equations I am trying to solve the following set of differential equations, I am trying to do it by the usual decoupling methods like adding the equations, subtracting etc which makes the process rather lengthy.Is there any other method, perhaps some transformation matrix which can simplify the problem.
\begin{equation}
 \left( \begin{array}{ccc}
-2\lambda +V(r) & \frac{\alpha}{2}\partial_r & 0 \\
\alpha(\partial_r+\frac{1}{r}) & V(r) & \alpha(\partial_r+\frac{1}{r})\\
0 & -\frac{\alpha}{2}\partial_r & 2\lambda+ V(r)
\end{array} \right) \left (\begin{array}{c}
\phi_1(r)\\
\phi_2(r)\\
\phi_3(r)
\end{array} \right)=\epsilon
\left( \begin{array}{c}
\phi_1(r)\\
\phi_2(r)\\
\phi_3(r)
\end{array} \right)
\end{equation}
 A: I have a solution for decoupling the equations, note that it is not through some transformation of the matrix. Rearranging your first and third equations gives
\begin{align}
\phi_1 &= \frac{1}{2 \lambda + \varepsilon - V(r)} \frac{\alpha}{2} \frac{d \phi_2}{d r}, \\
\phi_3 &= \frac{1}{2 \lambda -\varepsilon + V(r)} \frac{\alpha}{2} \frac{d \phi_2}{d r}.
\end{align}
And rearranging your second equation allows it to be written as 
\begin{align}
\frac{d}{dr} \big[ r(\phi_1 + \phi_3) \big] + \left( \frac{V(r) - \varepsilon}{\alpha} \right) r \phi_2 = 0.
\end{align}
Next I substitute the expressions given above for $\phi_1$ and $\phi_3$ into this equation, this yields the following second order ODE for the quantity $\phi_2(r)$
\begin{align}
\frac{d}{dr} \left(r \beta(r) \frac{d \phi_2}{dr}  \right) + \left( \frac{V(r) - \varepsilon}{\alpha^2} \right) r \phi_2 = 0,
\end{align}
where the function $ \beta(r) := \dfrac{2 \lambda}{[2 \lambda + \varepsilon - V(r)][2 \lambda - \varepsilon + V(r)]}$ has been defined for convenience. 
