Solution of the Ordinary Differential EquationsSolution of the Ordinary Differential Equations
The ordinary differential equation in time is an old friend.
The Differential Equation in t
Clear[w]Its solution istimeDiffEqn2
Clear[wSolution,t]We can satisfy the initial condition, du/dt (r,0) = 0 by settingwSolution[t_] := c1 Sin[c k t] + c2 Cos[c k t]
initialVelocity = wSolution'[0] == 0
We know that neither the k's nor c are zero, so we must set c1 = 0. So now
Clear[wSolution,t]Substituting the values of k we found in the previous section gives us a list of solutions to the wave equation, w[t] v[r].wSolution[t_] := c2 Cos[c k t]
solutionList = Thread[Times[Array[c,Length[j0zeroes]],wSolution[t]/c2 BesselJ[0,k r] /. j0zeroes]]
Each of these solutions satisfies the boundary conditions u(0,t) bounded and u(1,t) = 0, and the initial condition du/dt (r,0) = 0.