3.14 How to find the degree of an ODE?

Degree of an ode is the degree of the highest derivative in the oder. This needs a helper function

(*This function thanks to  Carl Woll, see https://mathematica.stackexchange.com/questions/151850/using-cases-and-when-to-make-input-a-list-or-not*) 
Clear["Global`*"]; 
getPatterns[expr_, pat_] :=   Last@Reap[expr /. a : pat :> Sow[a], _, Sequence @@ #2 &];
 

And now do

getDegreeOfOde[ode_,y_,x_]:=Module[{maxDer,p,d,der}, 
    der    = getPatterns[ode,Power[Derivative[_.][y][x],_.]] ; 
    p      = Flatten[Internal`ProcessEquations`DifferentialOrder[#,{x},{y}]&/@der]; 
    maxDer = Extract[der,Position[p,Max[p]]]; 
    Abs[First[maxDer/.(Derivative[_.][y][x])^(d_.):>d]] 
]
 

Examples of usage

ode=y'''[x]+y''[x]^4+3*y'[x]^8+3*y[x]^8==0; 
getDegreeOfOde[ode,y,x] 
   (*1*) 
 
ode=y''[x]^4+3*y'[x]^8+3*y[x]^8==0; 
getDegreeOfOde[ode,y,x] 
   (*4*) 
 
ode=y''[x]+3*y'[x]^8+3*y[x]^8==0; 
getDegreeOfOde[ode,y,x] 
   (*1*)