Problem 2

Horizontal

Clear["Global`*"]

(* To start, we'll zero gravity  *)
eq1= m v'[t] == m g - b v[t] - c v[t]^2 /.{g->0}

bc=v[0]==v0  (* Boundary Conditions *)

dsol1=DSolve[{eq1,bc} ,v[t],t][[1]] //FullSimplify

(* Look at ONLY LINEAR term *)
dsol2=DSolve[{eq1,bc} /.{c->0},v[t],t][[1]]

(* Look at ONLY QUADRATIC term *)
dsol3=DSolve[{eq1,bc} /.{b->0},v[t],t][[1]]

v[t] /.dsol1

v[t] /.dsol2

v[t] /.dsol3

(* Pick some values *)
values={v0->1, m->1};

v[t] /.dsol1 /.values

tmp1=v[t] /.{dsol1,dsol2,dsol3} /.values

Plot[tmp1 /.{b->1, c->1} //Evaluate,{t,0,10},
PlotStyle->{Red,Green,Blue},PlotLegends->LineLegend[{Red,Green,Blue},{"Gen","c=0","b=0"}]]

fun[b_,c_]=v[t] /.{dsol1,dsol2,dsol3} /.values

?Manipulate

Manipulate[
Plot[fun[b,c] //Evaluate,{t,0,10},
PlotStyle->{{Thickness[0.015],Red},Green,Blue},
PlotLabel->{"b=",b, "   c= ",c},
PlotLegends->LineLegend[{Red,Green,Blue},{"Both","Linear","Quadratic"}],
PlotRange->{Automatic,{0,1}}
]
,{{b,1,"Linear"},0.1,2,0.1},{{c,1,"Quadratic"},0.1,2,0.1}]