I would like to use DSolve (or NDSolve) to verify that the solution to the ODE problem
-4(v''[t]+(2/t)v'[t])-2*v[t]*Log[v[t]]-(3+(3/2)Log[4 Pi])*v[t]==0,
for $t\geq 0$ with conditions $\lim_{t\to \infty}v(t)=0$ and $v'(0)=0$ is given by the Gaussian
v[t]=(4 Pi)^(-3/4)*Exp[-t^2/8].
I am able to verify this by hand, but am having trouble using Mathematica to verify it. I would like to use Mathematica to solve this (simple) differential equation, so that I can later on modify some terms in the ODE to see how the solution changes.
Perhaps I am making a foolish mistake. I have also tried using NDSolve, but did not obtain the correct solution. I would appreciate any tips. Below you can find my code as well as the picture of the error messages. Thanks for your help.
sol=DSolve[{-4(v''[t]+(2/t)v'[t])-2*v[t]*Log[v[t]]
-(3+(3/2)Log[4 Pi])*v[t]==0,v[Infinity]==0,v'[0]==0},v[t],t]
Plot[Evaluate[v[t] /. sol], {t, 0, 10}, PlotRange -> All]
Edit: Changed r
to t
to correct a typo. The output is still the same as before.
DSolve
cannot solve for the general solution, which means there's little hope to getDSolve
to do it on its own. The particular solution for the BVP might be a special case that someone was able to derive.NDSolve
doesn't deal with infinity, so you have problems at both boundary conditions. You could use engineer's approximations, e.g.10^-8
for0
, and10
or100
or whatever for infinity (depends on scale). Still seems difficult. Maybe do shooting manually. $\endgroup$