How to solve the second order differential equation

Question

I've been trying to do code this for a two or so hours, and I can't seem to do it. Please help..

I am trying to solve the following second-order differential equation:

a(x'[t])^2 + b + c x[t] - m x''[t] = 0

where

{a = 0.00713, b = 0.0175, c = -12.141, m = 0.0044}
x[0]=0, x'[0]=0, x[0.233]=2

I want the answer to be in terms of

x'[t] = [Formula]

I'm expecting the x'[t] graph to be a sort of logarithmic function shaped.

Could anyone code this for me? (I'm currently stumped)

You've been explained how the to use the functions on Mathematica. As to why your differential equation is wrong is off topic here. Your differential equation is essentially a negatively damped harmonic oscillator, the form is correct for the differential equation you've got. — Feyre, Jan 07 '17 at 14:25
In both questions, you didn't provide the TWO initial conditions. Could you? — zhk, Jan 07 '17 at 14:37
By two initial conditions, is this sufficient? 'x[0]=0, x[0.233]=2, x'[0]=0' (sorry, I really don't know much about maths & mathematica yet..) — John, Jan 07 '17 at 14:40
John that's three conditions, and there is no solution to the equation which meets all three. — Simon Woods, Jan 07 '17 at 14:45
Hmm... the x[0.233]=2 was found experimentally, so I guess that can be ignored in this equation.
Then the two initial conditions become x[0]=0, x'[0]=0 — John, Jan 07 '17 at 14:46

score 1 · Answer 1 · edited Apr 13 '17 at 12:56

This is non-linear, and Mathematica can't solve it. But you can try Series solution. Using Michael E2 method in solving-an-ode-in-power-series

ClearAll[a, x, c, t, b, m];
eq = a (x'[t])^2 + b + c x[t] - m x''[t] == 0
Normal@seriesDSolve[a (x'[t])^2 + b + c x[t] - m x''[t] == 0, 
  x, {t, 0, 3}, {x[0] -> A0, x'[0] -> B0} (*A0,B0 are initial conditions*)

The answer is

Mathematica graphics

This matches Maples's answer for series

restart;
eq := a *( diff(x(t),t))^2 + b + c * x(t) - m* diff(x(t),t$2);
Order:=4;
dsolve({eq=0,x(0)=A0,D(x)(0)=B0},x(t),'series');

Mathematica graphics

Now you can plugin your parameters values in the solution above.

score 1 · Accepted Answer · edited Apr 13 '17 at 12:55

In response to your first question, Feyre has explained in great detail the procedure to get whatever you wanted. But it seems, you are still unable to do it. And I suspect this problem arise when you were trying to plot x'[t] from NDSolve. This issue was also addressed by Feyre in his comments and suggested to use NDSolveValue.

Eqn = a*(x'[t])^2 + b + c *x[t] - m *x''[t] == 0; 
a = 0.00713; b = 0.0175; c = -12.141; m = 0.044;
sol = NDSolveValue[ {Eqn, x[0] == 0, x'[0] == 0}, x[t], {t, 0, 2}]
Show[Plot[Evaluate[sol], {t, 0, 2}, PlotStyle -> {Red}, 
  PlotLegends -> {"x[t]"}], 
 Plot[Evaluate[D[sol, t]], {t, 0, 2}, PlotStyle -> {Green}, 
  PlotLegends -> {"x'[t]"}], PlotRange -> All]

Now to see whether the Mathematica agree with the so-called experimental value you stated,

CEData = (Evaluate[sol] /. t -> 0.233)

0.00251695

which is far from close to the data you provided. So, I also checked with Maple,

restart;with(plots):
eq := a *( diff(x(t),t))^2 + b + c * x(t) - m* diff(x(t),t$2);
a:= 0.00713; b := 0.0175; c := -12.141; m := 0.044;
ics:=D(x)(0)=0,x(0)=0;
sol:=dsolve({eq,ics},numeric);
odeplot(sol,[[t,diff(x(t),t)],[t,x(t)]],0..2,color=
[red,green],axes=boxed,legend=["x'(t)","x(t)"]);

sol(0.233)

0.00251697716038847

I think you should recheck your differential equation along with the parameter values.

If you are interested in finding the exact solution then you should follow Nasser's answer?

How to solve the second order differential equation

2 Answers2