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plot({ exp(-x^2/4)/sqrt(4*Pi),
   exp(-x^2/(4*2))/sqrt(4*Pi*2),
   exp(-x^2/(4*4))/sqrt(4*Pi*4)},
   x=-10..10,color=BLACK);

              Exercises/Experiments
1.
with(stats): with(plots):
for i from 1 to 100 do
   count[i]:=0; od: # initialize a counter
N:=rand(0..1): # random integer 0/1
particles:=500; steps:=40;
for m from 1 to particles do
place:=sum('2*N(p)-1','p'=1..steps)+steps:
count[place]:=count[place]+1: # record endpt
od:
# histogram the endpoints
ranges:=[seq(-steps/2+2*(i-1)..-steps/2+2*i,i=1..steps/2)];
movement:=[seq(count.(20+2*j),j=1..20)];
diffusion:=[seq(Weight(ranges[i],movement[i]),i=1..20)];
statplots[histogram](diffusion);

1a.
plot({ exp(-x^2/4)/sqrt(4*Pi),
   exp(-x^2/(4*2))/sqrt(4*Pi*2),
exp(-x^2/(4*3))/sqrt(4*Pi*3)},x=-10..10);

1b.
u:=(t,x)-> exp(-x^2/(4*t))/sqrt(4*Pi*t);
diff(u(t,x),t)-diff(u(t,x),x,x);
simplify(%);

1c.
diff(U(t,x,y),t) = diff(U(t,x,y),x,x)+diff(U(t,x,y),y,y);
simplify(%);

U:=(t,x,y)->exp(-(x^2+y^2)/(4*t))/t;
diff(U(t,x,y),t) - diff(U(t,x,y),x,x)-diff(U(t,x,y),y,y);
simplify(%);

1d.
plot({ exp(-x^2/4)/sqrt(4*Pi),exp(-x^2/(4*2))/sqrt(4*Pi*2),
  exp(-x^2/(4*3))/sqrt(4*Pi*3)},x=-10..10);
with(plots):
animate(exp(-x^2/(4*t))/sqrt(4*Pi*t),x=-10..10,t=0.1..5);
animate3d(exp(-(x^2+y^2)/(4*t))/t,x=-1..1,y=-1..1,t=0.1..0.5);

2.
c:=(t,x)->exp(-t)*sin(x);

diff(c(t,x),t)-diff(c(t,x),x,x);

with(plots):
animate(c(t,x),x=0..2*Pi,t=0..2);