equation solving - Using DSolve to solve for x[t,parameter]

I'm trying to solve an ODE with two independent variables (a cannon firing from a cliff incorporating wind resistance dependent on velocity). I've tried the following for the x-component:

NDSolve[{x''[t, θ] == -0.2*x'[t, θ]/2.30, x'[0, θ] == 10.8*Cos[θ], x[0, θ] == 0},
  x[t, θ], t]

Assuming my algebra is correct, how do I properly phrase this input to give me x[t, θ]?

Answer

If :

sol = DSolve[{D[x[t, th], {t, 2}] == -0.2*D[x[t, th], t]/2.30, 
 Derivative[1, 0][x][0, th] == 10.8*Cos[th], x[0, th] == 0}, x[t, th], t];

then you can use the solution as :

Plot3D[x[t, th] /. sol, {t, 0, 10}, {th, -Pi, Pi}]

enter image description here

Some checks :

x[t, th] /. First[sol] /. t -> 0
(* 0. *)

Simplify[D[sol[[1, 1, 2]], t] /. t -> 0]

(* 10.8 Cos[th] *)

A more general and robust way to keep the solution for future use is to define :

Remove[sol]
sol[t_, th_] = DSolve[{D[x[t, th], {t, 2}] == -0.2*D[x[t, th], t]/2.30, 
 Derivative[1, 0][x][0, th] == 10.8*Cos[th], x[0, th] == 0}, x[t, th], t][[1, 1, 2]];

which you can now use as any other function. For instance :

Manipulate[Plot[sol[t, th], {t, 0, 10}], {th, 0, Pi/2}]

Comments

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functions - Get leading series expansion term?

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What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...

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