Solution of the Ordinary Differential Equations

Solution of the Ordinary Differential Equations

   
The Differential Equation in t
The ordinary differential equation in time is an old friend.

In[32]:=

Clear[w]

timeDiffEqn2


Out[33]=

             2  2

w''[t] == -(c  k  w[t])
Its solution is

In[34]:=

Clear[wSolution,t]

wSolution[t_] := c1 Sin[c k t] + c2 Cos[c k t]
We can satisfy the initial condition, du/dt (r,0) = 0 by setting

In[36]:=

initialVelocity = wSolution'[0] == 0


Out[36]=

c c1 k == 0
We know that neither the k's nor c are zero, so we must set c1 = 0. So now

In[37]:=

Clear[wSolution,t]

wSolution[t_] := c2 Cos[c k 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]. 

In[39]:=

solutionList = Thread[Times[Array[c,Length[j0zeroes]],

   wSolution[t]/c2 BesselJ[0,k r] /. j0zeroes]]



Out[39]=

{BesselJ[0, 2.40483 r] c[1] Cos[2.40483 c t], 

 

  BesselJ[0, 5.52008 r] c[2] Cos[5.52008 c t], 

 

  BesselJ[0, 8.65373 r] c[3] Cos[8.65373 c t], 

 

  BesselJ[0, 11.7915 r] c[4] Cos[11.7915 c t]}
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.

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