# Entering complicated differential equation in mathematica

I am trying to enter a differential equation in mathematica. Example of such an equation is $$\bigg[\bigg(\frac{\partial{}}{\partial{t}}+\omega_\eta\bigg)\left(\frac{\partial{}}{\partial{t}}+\omega_\nu\right)+\omega_M^2\bigg]^2\left(\frac{\partial}{\partial{t}}+\omega_\kappa\right)\hat{u}_y=-4\omega_C^2\left(\frac{\partial{}}{\partial{t}}+\omega_\eta\right)^2\left(\frac{\partial}{\partial{t}}+\omega_\kappa\right)\hat{u}_y-\omega_A^2 \bigg[\left(\frac{\partial{}}{\partial{t}}+\omega_\nu\right)\left(\frac{\partial{}}{\partial{t}}+\omega_\eta\right)+\omega_M^2\bigg]\left(\frac{\partial{}}{\partial{t}}+\omega_\eta\right)\hat{u}_y,$$ where the unknown $$\hat{u}_y$$ is a function of time and space. How can it be entered into mathematica without expanding it completely on a paper and then writing the whole thing in mathematica? This is what I did

Le[f_] := \!$$\*SubscriptBox[\(\[PartialD]$$, $$t$$]f\) + \[Omega]e f;
Ln[f_] := \!$$\*SubscriptBox[\(\[PartialD]$$, $$t$$]f\) + \[Omega]n f;
Lk[f_] := \!$$\*SubscriptBox[\(\[PartialD]$$, $$t$$]f\) + \[Omega]k f;
Lg[f_] := Le[Ln[f]] + \[Omega]M^2 f;
Lgg[f_] := Lg[Lg[Lk[f]]];
Lek[f_] := Le[Le[Lk[f]]];
Lge[f_] := Lg[Le[f]];


and then

Collect[Simplify[
Lgg[u[t]] + 4 \[Omega]o^2 Lek[u[t]] + \[Omega]A^2 Lge[u[t]]], {
\!$$\*SuperscriptBox[\(u$$,
TagBox[
RowBox[{"(", "5", ")"}],
Derivative],
MultilineFunction->None]\)[t],
\!$$\*SuperscriptBox[\(u$$,
TagBox[
RowBox[{"(", "4", ")"}],
Derivative],
MultilineFunction->None]\)[t],
\!$$\*SuperscriptBox[\(u$$,
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[t],
\!$$\*SuperscriptBox[\(u$$,
TagBox[
RowBox[{"(", "2", ")"}],
Derivative],
MultilineFunction->None]\)[t],
\!$$\*SuperscriptBox[\(u$$,
TagBox[
RowBox[{"(", "1", ")"}],
Derivative],
MultilineFunction->None]\)[t],
\!$$\*SuperscriptBox[\(u$$,
TagBox[
RowBox[{"(", "0", ")"}],
Derivative],
MultilineFunction->None]\)[t]}]


(((der + w1) (der + w2) + w3^2)^2 (der + w4) // Expand ) /.