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MathGroup Archive 2001

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Re: 2nd order differential equation help

  • To: mathgroup at smc.vnet.net
  • Subject: [mg31525] Re: 2nd order differential equation help
  • From: BobHanlon at aol.com
  • Date: Fri, 9 Nov 2001 06:13:28 -0500 (EST)
  • Approved: Steven M. Christensen <steve@smc.vnet.net>, Moderator
  • Sender: owner-wri-mathgroup at wolfram.com

In a message dated 2001/11/7 7:48:34 AM, peterangelo at mindspring.com writes:

>actly how do I get this into a form Mathamatica will accept?  It has
>been a number of years since enrolling in diffeq, however I think I am
>close to correct but would appreciate help.
>
>the problem;
>
>Solve the differential equation:
>       
>g''(t) + 0.1 g'(t) + g(t) = 0 
>
>where the previous equation uses conventional mathematical notation. 
>You must transfrom the equation and initial conditions into
>Mathematica syntax.
>
>for t between 0 and 10 Pi
>
>subject to the initial conditions
>   g(0) = 0,
>   g'(0) = 1
>
>Plot your results.
>       
>My attempt is as follows:
>
>
>Clear[g]
>
>Clear[eqn]
>
>eqn[t_] = g''[t] + 0.1g'[t] + g[t] == 0
>
>
>Clear[eqn1]
>
>eqn1[t_] = g'[t] + 0.1g[t] + ?g[t] == 0
>
>solution = DSolve[{eqn[t], eqn1[t]}, g'[0] == 1, 
>g[0] == 0, g[t], {t, 0, 10\[Pi]}]
>
>DSolve::dsvar: g[0]==0 cannot be used as a variable.
>
>what am I dont doing right?  any suggestions helpful
>

Clear[g];

g[t] /. DSolve[{g''[t]+g'[t]/10+g[t]==0, 
        g[0] == 0, g'[0] == 1}, 
      g[t], t][[1]]

(10*I*(E^((1/20)*(-1 - I*Sqrt[399])*
      t) - E^((1/20)*(-1 + 
       I*Sqrt[399])*t)))/Sqrt[399]

ExpToTrig [%]

(10*I*Cosh[(1/20)*(-1 - 
       I*Sqrt[399])*t])/Sqrt[399] - 
  (10*I*Cosh[(1/20)*(-1 + 
       I*Sqrt[399])*t])/Sqrt[399] + 
  (10*I*Sinh[(1/20)*(-1 - 
       I*Sqrt[399])*t])/Sqrt[399] - 
  (10*I*Sinh[(1/20)*(-1 + 
       I*Sqrt[399])*t])/Sqrt[399]

FullSimplify[%]

(20*Sin[(Sqrt[399]*t)/20])/
  (E^(t/20)*Sqrt[399])

Consolidating these steps into a definition for g[t]

g[t_] := 
    Evaluate[FullSimplify[
        ExpToTrig[g[t] /. 
            DSolve[{g''[t]+g'[t]/10+g[t]==0, 
                  g[0] == 0, g'[0] == 1}, 
                g[t], t][[1]]]]];

Checking

g''[t]+g'[t]/10+g[t] == 0 // Simplify

True

Plot[g[t], {t, 0, 10Pi}];


Bob Hanlon
Chantilly, VA  USA


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