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