Re: Mathematica won't solve simple diff. eqn.--Correction

• To: mathgroup at smc.vnet.net
• Subject: [mg24771] Re: Mathematica won't solve simple diff. eqn.--Correction
• From: "Edgardo S. Cheb-Terrab" <ecterrab at daisy.uwaterloo.ca>
• Date: Thu, 10 Aug 2000 00:31:52 -0400 (EDT)
• References: <8mdlt6\$5jt@smc.vnet.net> <8mr0t8\$1kr@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```"Christopher R. Carlen" <crcarle at sandia.gov> wrote in message
news:8mr0t8\$1kr at smc.vnet.net...
> ...
> ...
> When I do:
>
> In:
>
> DSolve[{-4 i1'[t] + 8 i2'[t] - 25 i1[t] + 20 i2[t] == -80 + 720 E^(-5 t),
> -4 i1'[t] + 8 i2'[t] - 10 i1[t] + 40 i2[t] == 640 E^(-5 t)}, {i1, i2}, t]
>
> Mathematica 4.0 simply outputs the DSolve statement with no result.
>
> When I do:
>
> In:
>
> i1 = 4 + 64 Exp[-5 t] - 68 Exp[-4 t]
> i2 = 1 - 52 Exp[-5t] + 51 Exp[-4 t]
>
> Simplify[ -4 D[i1, t] + 8 D[i2, t] - 25 i1 + 20 i2 == -80 + 720 Exp[-5 t] ]
> Simplify[ -4 D[i1, t] + 8 D[i2, t] - 10 i1 + 40 i2 == 640 Exp[-5 t] ]
>
> Out:
>
> true
> true
>
> indicates that the solutions are valid.
>
> The question is then:
>
> Why can't Mathematica solve the system?
>
> _______________________
> Christopher R. Carlen
> Sr. Laser/Optical Tech.
> Sandia National Labs
>
=====================================================================
Hi,
First of all, your are right: your system indeed has a solution. I'm rewriting
it with x = i1 and y = i2:

In[1]:= {-4 x'[t] + 8 y'[t] - 25 x[t] + 20 y[t] == -80 + 720 E^(-5 t),
-4 x'[t] + 8 y'[t] - 10 x[t] + 40 y[t] == 640 E^(-5 t)} ;

A solution for this system, a bit more general than what you are showing,
in terms of an arbitrary constant C[1], is given by:

In[2]:= {x[t] == 4+64/Exp[t]^5+C[1]/Exp[t]^4,
y[t] == 1-52/Exp[t]^5-3/4*C[1]/Exp[t]^4}

64    C[1]                52    3 C[1]
Out[2]= {x[t] == 4 + ---- + ----, y[t] == 1 - ---- - ------}
5 t    4 t               5 t      4 t
E      E                 E      4 E

as it is easy to verify. Now, why is DSolve failing? I believe that is
because your system is not really a system of 2 first order ODEs but
actually a single first order ODE plus an algebraic equation in disguised
form. In other words: if you compute the integrability conditions implied
by your system In[1], you will see it can be rewritten as

In[3]:= {y[t] == -3/4*x[t]+4-4/Exp[t]^5, x'[t] == -4*x[t]+16-64/Exp[t]^5}

4     3 x[t]                 64
Out[3]=  {y[t] == 4 - ---- - ------, x'[t] == 16 - ---- - 4 x[t]}
5 t     4                    5 t
E                            E

That is: involving the first derivative of x[t] and no derivatives of
y[t]. That is also why the solution depends on a single arbitrary constant
(C[1] in In[2]) instead of two of them.

Hope this is of use.

Edgardo
___________________________________________________________________________
Edgardo S. Cheb-Terrab                     http://lie.uwaterloo.ca/ecterrab
Theoretical Physics Department                                  UERJ,Brazil
Centre for Experimental and Constructive Mathematics            SFU, Canada

```

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