ClearAll["Global`*"]

Examination of solution to ODE Example Prob 8-1-7

For this problem, a mass is accelerated by a constant force. It starts at rest at the origin.
Eventually, we’ll plug in numerical values:  m = 0.1 kg and F = 100 N.
If we knew nothing about relativity, we would suspect that the answers obeyed the simple constant acceleration laws from Physics I:

a = (F/m);x1[t_]=(1/2) a (t^2);v1[t_]=a t;        (* "Newtonian"*)

However, what we actually found was this:

x2[t_]=((c/F)) (Sqrt[(m^2)(c^2)+(F^2)(t^2) ]) - ((m (c^2))/F);          (* "Relativistic": x2 is the "true" solution we found by hand*)
v2[t_]=(Sqrt[(((c^2)(F^2)(t^2))/((c^2)(m^2)+(F^2)(t^2))) ]);

Let’s look at these two solutions over a modest time period to see whether they agree

m = 0.1;F = 100.0; c = 2.998 (10^8);tmax = 1 (10^4);
(* Red is Newtonian, Blue is relativistic *)
plot1=Plot[x1[t],{t,0,tmax},PlotStyle->Directive[RGBColor[1,0,0],AbsoluteThickness[6]]];
plot2=Plot[x2[t],{t,0,tmax},PlotStyle->Directive[RGBColor[0,0,1],AbsoluteThickness[2]]];
Show[plot1,plot2,ImageSize->{600,300},AspectRatio->Full] (* Plot Positions *)

plot3=Plot[v1[t],{t,0,tmax},PlotStyle->Directive[RGBColor[1,0,0],AbsoluteThickness[6]]];
plot4=Plot[v2[t],{t,0,tmax},PlotStyle->Directive[RGBColor[0,0,1],AbsoluteThickness[2]]];
Show[plot3,plot4,ImageSize->{600,300},AspectRatio->Full] (* Plot velocities *)

Well, those look pretty good. But, what if we compare the solutions over a long time period?

m = 0.1;F = 100.0; c = 2.998 (10^8);tmax = 1 (10^6);
plot1=Plot[x1[t],{t,0,tmax},PlotStyle->Directive[RGBColor[1,0,0],AbsoluteThickness[6]]];
plot2=Plot[x2[t],{t,0,tmax},PlotStyle->Directive[RGBColor[0,0,1],AbsoluteThickness[2]]];
plot3=Plot[v1[t],{t,0,tmax},PlotStyle->Directive[RGBColor[1,0,0],AbsoluteThickness[6]]];
plot4=Plot[v2[t],{t,0,tmax},PlotStyle->Directive[RGBColor[0,0,1],AbsoluteThickness[2]]];
Show[plot1,plot2,ImageSize->{600,300},AspectRatio->Full]
Show[plot3,plot4,ImageSize->{600,300},AspectRatio->Full]

Here, the blue line is the correct line... you can see that in the second plot (velocities), the red (constant acceleration equation) line shows that the speed of the particle has significantly surpassed the speed of light!