# Reducing Higher Order Coupled Ordinary Differential Equations

I am looking to reduce fifth order linear coupled ordinary differential equations into first order so that I could apply the math already developed for linear equations. I am not sure if there is a way to reduce them using Mathematica. The equations are

``````
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));

eq1 = Collect(Simplify(
Lgg(uz(t)) + (Omega)C^2 Lek(uz(t)) + (Omega)A^2 Lge(uz(t))), {

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "5", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "4", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "3", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "2", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "1", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "0", ")"}),
Derivative),
MultilineFunction->None))(t)}) /. {(Omega)n -> 0, (Omega)k ->
0} /. (Omega)k -> 0

eq2 = Collect(Simplify(
Lgg(ux(t)) + (Omega)C^2 Lek(ux(t)) -
2 I (CapitalOmega) g (Beta) ky/k^2 Lek(
uz(t)) + (Alpha) g (Beta) kx kz/k^2 Lg(Le(uz(t)))), {

!(*SuperscriptBox((ux),
TagBox(
RowBox({"(", "5", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((ux),
TagBox(
RowBox({"(", "4", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((ux),
TagBox(
RowBox({"(", "3", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((ux),
TagBox(
RowBox({"(", "2", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((ux),
TagBox(
RowBox({"(", "1", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((ux),
TagBox(
RowBox({"(", "0", ")"}),
Derivative),
MultilineFunction->None))(t),
!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "5", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "4", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "3", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "2", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "1", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "0", ")"}),
Derivative),
MultilineFunction->None))(t)}) /. {(Omega)n -> 0, (Omega)k ->
0} /. (Omega)k -> 0

eq3 = Collect(Simplify(
Lgg(uy(t)) + (Omega)C^2 Lek(uy(t)) +
2 I (CapitalOmega) g (Beta) kx/k^2 Lek(
uz(t)) - (Alpha) g (Beta) ky kz/k^2 Lg(Le(uz(t)))), {

!(*SuperscriptBox((uy),
TagBox(
RowBox({"(", "5", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uy),
TagBox(
RowBox({"(", "4", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uy),
TagBox(
RowBox({"(", "3", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uy),
TagBox(
RowBox({"(", "2", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uy),
TagBox(
RowBox({"(", "1", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uy),
TagBox(
RowBox({"(", "0", ")"}),
Derivative),
MultilineFunction->None))(t),
!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "5", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "4", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "3", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "2", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "1", ")"}),
Derivative),
MultilineFunction->None))(t),

!(*SuperscriptBox((uz),
TagBox(
RowBox({"(", "0", ")"}),
Derivative),
MultilineFunction->None))(t)}) /. {(Omega)n -> 0, (Omega)k ->
0} /. (Omega)k -> 0
``````

I want to reduce them into a system of equations to be written in a matrix form. Although I have used partials, they are ordinary differential equations in t.

Or if there is any other way to solve it using mathematica, please let me know.

Thank you.