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My equation is

[{x[t]*x'[t])'-(F/m)+(b/m)*x(t)*x'(t)==0},x(0)=0,x'(0)=0]

It is a form of Newtons momentum equation, but I am having a lot of trouble solving this in Mathematica. Is DSolve the correct method to go about this?

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  • $\begingroup$ Please describe the underlying mechanical problem a little bit more. $\endgroup$ Commented Jan 18, 2019 at 17:16
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    $\begingroup$ Please see The Four Kinds of Bracketing in the Wolfram Language $\endgroup$
    – Bob Hanlon
    Commented Jan 18, 2019 at 17:21
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    $\begingroup$ To answer your question, DSolve is the first function to try in this case. $\endgroup$
    – bbgodfrey
    Commented Jan 19, 2019 at 16:32

2 Answers 2

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Are you sure about the initial conditions?

DSolve is the right method. Observing that only the product x[t]x'[t] occurs the substitution x[t]x'[t]->u[t] gives a first order ode which can be solved for u[t]

U = DSolveValue[ {D[u[t] , t] - (F/m) + (b/m)*u[t] == 0  }, u, t]
(*Function[{t}, F/b + E^(-((b t)/m)) C[1]]*)

Resubstituting u[t] gives the solution x[t]

X = DSolveValue[{x[t] x'[t] == U[t]   }, x, t]
(*Function[{t}, -((Sqrt[2] Sqrt[F t - E^(-((b t)/m)) m C[1] + b C[2]])/Sqrt[b])]*)

That is the general solution of your problem! The parameters C[1],C[2] must be choosen to fullfill the initial conditions

{X[0], X'[0]}
(*{-((Sqrt[2] Sqrt[-m C[1] + b C[2]])/Sqrt[b]), -((F + b C[1])/(Sqrt[2] Sqrt[b] Sqrt[-m C[1] + b C[2]]))}*)

Unfortunately it is not possible to adapt your initial conditions x[0]==0,x'[0]==0

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Please read these introductions before continuing. What appears to be the function you want is DSolve[{\!\( \*SubscriptBox[\(\[PartialD]\), \(t\)]\((x[t]*\(x'\)[t])\)\) - (F/ m) + (b/m)*x[t]*x'[t] == 0, x[0] == 0, x'[0] == 0}, x[t], t], which has no solutions for your initial conditions.

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