In[205]:= eq1 =
Assuming[\[Lambda] \[Element] Reals,
DSolve[{(\[Phi]^\[Prime]\[Prime])[r1]/\[Phi][r1] == -\[Lambda]^2,
k \[Phi][0] == Derivative[1][\[Phi]][0],
k \[Phi][l] == Derivative[1][\[Phi]][l]}, \[Phi][r1], r1]]
Out[205]= {{\[Phi][r1] -> 0}}
In[206]:= eq2 =
Assuming[\[Lambda] \[Element] Reals,
DSolve[{(\[Phi]^\[Prime]\[Prime])[r1]/\[Phi][r1] == \[Lambda]^2,
k \[Phi][0] == Derivative[1][\[Phi]][0],
k \[Phi][l] == Derivative[1][\[Phi]][l]}, \[Phi][r1], r1]]
Out[206]= {{\[Phi][r1] -> 0}}
In[207]:= eq3 =
DSolve[{(\[Phi]^\[Prime]\[Prime])[r1]/\[Phi][r1] == -\[Lambda]^2,
k \[Phi][0] == Derivative[1][\[Phi]][0],
k \[Phi][l] == Derivative[1][\[Phi]][l]}, \[Phi][r1], r1,
Assumptions -> \[Lambda] \[Element] Reals]
Out[207]= {{\[Phi][r1] -> 0}}
In[208]:= eq4 =
DSolve[{(\[Phi]^\[Prime]\[Prime])[r1]/\[Phi][r1] == \[Lambda]^2,
k \[Phi][0] == Derivative[1][\[Phi]][0],
k \[Phi][l] == Derivative[1][\[Phi]][l]}, \[Phi][r1], r1,
Assumptions -> \[Lambda] \[Element] Reals]
Out[208]= {{\[Phi][r1] -> 0}}
In the 4 pieces of codes above, there should appear a solution as c1cos(lambda x) + c2sin(lambda x).