# DSolve and Erf trouble

```hi Guru's,

two questions in a one-line problem :

1/ First part:

An inconsistency came up in a differential equation :

sol=Flatten@DSolve[{q'[t]== -a q[t] r[t], r'[t]== -a q[t] r[t]+e, q[0]==
Q, r[0]== R},{q[t],r[t]},t]
Part::partw: Part 2 of r'[q] does not exist. Part::partw: Part 2 of
r'[q] does not exist. Out[90]=
DSolve[{q'[t] == -(a q[t] r[t]), r'[t] == e - a q[t] r[t], q[0] == Q,
r[0] == R}, {q[t], r[t]}, t]

... meaning : "I don't work on stuff like this ..." (:-(( ...
what's  r'[q] got to do with it?

but with a little human intervention,  sol=Flatten@DSolve[{q'[t]== -a
q[t] r[t], r'[t]== q'[t]+e, q[0]== Q, r[0]== R},{q[t],r[t]},t]

Solve::ifun: Inverse functions are being used by Solve, so some
solutions may not be found.
Out[1]=
{r[t] -> R + e*t - q[0] + q[t],
q[t] -> (-2*Sqrt[e])/
(E^((a*t*(e*t + 2*(R - q[0])))/2)*
(Sqrt[e]*(-2/Q + (Sqrt[a]*E^((a*(R - q[0])^2)/(2*e))*Sqrt[2*Pi]*
Erf[(Sqrt[a]*(R - q[0]))/(Sqrt[2]*Sqrt[e])])/Sqrt[e]) -
Sqrt[a]*E^((a*(R - q[0])^2)/(2*e))*Sqrt[2*Pi]*
Erf[(Sqrt[a]*(R + e*t - q[0]))/(Sqrt[2]*Sqrt[e])]))}

... meaning : "got it right this time, that's what I call input"  (:-))
...            but what's q[0] doing in there? I told you it was to be
"Q".

2/ Second part.

Now this solution contains a very large Exponential, multiplied with a
very small difference of two Erf functions. That's asking for trouble
in Mathematica 3.0.0 since numerical float of Erf[ large ] goes
haywire. How can it be reduced to a form appropriate for numerical
evaluation?

a=.02;Q=50;R=.001;e=1.;
Plot[{qq[t],rr[t]},{t,0,100},PlotRange->All]

works ok, but ...

In[104]:=Clear[a,e,R,Q];a=.05;Q=50;R=0.001;e=1.;
In[105]:=Table[qq[t],{t,0,20}]

Out[105]=
{50., 594.0556521297816, 6713.817568919678, 72176.73228552992,
738091.4452270053, 7.179734807316404*^6, 6.643423078153096*^7,
5.847371602002258*^8, 4.895699508759986*^9, 3.899007663442238*^10,
2.953783770341961*^11, 2.128573185928179*^12, 1.459095606974182*^13,
44.06762248565081, 39.04696606830096, 35.83635349893076,
34.54389468503202, 33.57633898133737, 32.60143669029274,
31.61976984484918, 30.63888981226333}

...  blows up.

how can I explore larger a-values?

wouter.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc@vandemoortele.be
eu000949@pophost.eunet.be

```

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