I think the problem can be solved by giving initial conditions.
In[2345]:= eq71 = DSolve[{f2[k, t] - Dp*w2[k, t] \[Eta][k] ==
\!\(\*SuperscriptBox[\(w2\), \*
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[k, t], f3[k, t] - Dp*w3[k, t] \[Eta][k] ==
\!\(\*SuperscriptBox[\(w3\), \*
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[k, t], w2[k, 0] == eq71,
w3[k, 0] == eq72}, {w2[k, t], w3[k, t]}, t] // Simplify // Normal
Out[2345]= {{w2[k, t] -> (1/((lh - rb) Sqrt[\[Eta][k]]))
E^(-Dp t \[Eta][
k]) (2 pinf (Sin[lh Sqrt[\[Eta][k]]] -
Sin[rb Sqrt[\[Eta][k]]]) + (-lh + rb) (\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(1\), \(0\)]\(\(
\*SuperscriptBox[\(E\), \(Dp\ K[1]\ \[Eta][k]\)]\ f2[k,
K[1]]\) \[DifferentialD]K[1]\)\)) Sqrt[\[Eta][
k]] + (lh - rb) (\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(1\), \(t\)]\(\(
\*SuperscriptBox[\(E\), \(Dp\ K[1]\ \[Eta][k]\)]\ f2[k,
K[1]]\) \[DifferentialD]K[1]\)\)) Sqrt[\[Eta][k]]),
w3[k, t] -> (1/((lh - rb) Sqrt[\[Eta][k]]))
E^(-Dp t \[Eta][
k]) (2 pinf (-Cos[lh Sqrt[\[Eta][k]]] +
Cos[rb Sqrt[\[Eta][k]]]) + (-lh + rb) (\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(1\), \(0\)]\(\(
\*SuperscriptBox[\(E\), \(Dp\ K[2]\ \[Eta][k]\)]\ f3[k,
K[2]]\) \[DifferentialD]K[2]\)\)) Sqrt[\[Eta][
k]] + (lh - rb) (\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(1\), \(t\)]\(\(
\*SuperscriptBox[\(E\), \(Dp\ K[2]\ \[Eta][k]\)]\ f3[k,
K[2]]\) \[DifferentialD]K[2]\)\)) Sqrt[\[Eta][k]])}}