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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]}]
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One possibility is to first replace the derivative by a dummy (e.g. "der"). Then do the expand. And finally replace the dummy by the derivative. Here, as an example, the first line of your equation:

(((der + w1) (der + w2) + w3^2)^2 (der + w4) // Expand ) /. 
 der^(n_ : 1) -> Derivative[n][u[t]]

unfortunately, this results in an ugly unformatted expression if pasted directly. I therefore paste a picture of the formatted expression: enter image description here

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