Re: Mathematica won't solve simple diff. eqn. system

*To*: mathgroup at smc.vnet.net*Subject*: [mg24727] Re: Mathematica won't solve simple diff. eqn. system*From*: changpoohbear at my-deja.com*Date*: Wed, 9 Aug 2000 02:31:26 -0400 (EDT)*References*: <8mdlt6$5jt@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Christopher, It appears that the general linear system that you are trying to solve is in the form of a i1'[t] + b i2'[t] + c i1[t] + d i2[t] ==0 e i1'[t] + f i2'[t] + g i1[t] + h i2[t] ==0 i1[0] == alpha i2[0] == beta Since this is an autonomous system, it also can be represented in matrix form as A i'[t] = B i[t] where A={{a,b},{e,f}}; B=-{{c,d},{g,h}}; i[t]={{i1[t]},{i2[t]}}; *IF* A is invertible, the solution can be given as i[t] = MatrixExp[Inverse[A].B].i[0] where i[0] = {{i1[0]},{i2[0]}} In your example given below, I don't believe the solutions given are correct ({{-4,8},{-4,8}} also is not invertible!). For your specific example, observe that Eliminate[{-4 i1'[t] + 8i2'[t] - 25i1[t] + 20 i2[t] == 0,-4i1'[t] + 8 i2'[t] - 10 i1[t] + 40 i2[t] == 0}, {i1'[t],i2'[t]}] yields 3 i1[t]==-4 i2[t] Using this fact, together with the fact that i1[0]=i2[0]=0, one should obtain i1[t]=i2[t]=0 I don't know why my version of Mathematica 3.0.x cannot realise this, and just returns its input as its output when using DSolve. <Shrug> I hope this helps! Mike In article <8mdlt6$5jt at smc.vnet.net>, "Christopher R. Carlen" <crcarle at sandia.gov> wrote: > Mathematica 4.0 and linear constant coefficient differential equations: > > I have the following system: > > -4 i1'[t] + 8 i2'[t] - 25 i1[t] + 20 i2[t] == 0 > -4 i1'[t] + 8 i2'[t] - 10 i1[t] + 40 i2[t] == 0 > i1[0]==0 > i2[0]==0 > > Which when I try to solve with DSolve, it fails. > > It seems any system of the form: > > y1'[t] + y2'[t] + C1 y1[t] + C2 y2[t] == 0 > y1'[t] + y2'[t] + C3 y1[t] + C4 y2[t] == 0 > > can't be solved. If the coefficients on y1' and y2' are not the same > between the two equations, then it can be solved. > > The problem is that there is a solution to the above system, which I > have verified. That solution is: > > i1[t_] = 4 + 64 E^(-5 t) - 68 E^(-4 t) > i2[t_] = 1 - 52 E^(-5 t) + 51 E^(-4 t) > > So the question is: If there is a solution (and not a very difficult > one) why can't Mathematica find it??? Is there some way to coerce Mathematica to > produce the equation, in both the symbolic and numerical situations? > > These types of systems arise frequently in the study of electronic > circuits. Numerical solvers like SPICE solve them without any > difficulty. I have struggled with getting Mathematica to solve them for a long > time. Sometimes I force a numerical solution by perturbing the > coefficients a bit, as long as the error is acceptible. > > But I would like to understand better what the hangup is. I have had a > diff. eqns. course, but haven't gone into systems yet. > > Thanks. > -- > _______________________ > Christopher R. Carlen > Sr. Laser/Optical Tech. > Sandia National Labs > > Sent via Deja.com http://www.deja.com/ Before you buy.