@acer  
restart;
simplify(solve({x=r*cos(theta), y=r*sin(theta), r>0}, {r,theta}, explicit))  assuming real;

@Ramakrishnan  I can not understand the reasons for the errors, because I do not have enough information. What is  BP  and what are the elements of the matrix  M1. Here's what I see when I try to run your code:

@Ramakrishnan 

The command  plot([[x1,y1],[x2,y2]], x=a..b)  just builds the segment that connects the points  [x1,y1]  and  [x2,y2] , regardless of the view  x=a..b

But if we want to continue this segment to the whole range a..b , then we need the equation of the corresponding straight line and  g  procedure provides us with this equation. x1, y1, x2, y2 are local variables in this procedure and their names do not matter. For example, g:=(a,b,c,d)->b+(d-b)*(x-a)/(c-a)  will be the same.

@aininabdul  Also you missed the multiplication sign after  exp((1/2)*x) . I did not notice this at first.

@Nemo_ 

Try 2 things:

1. Execute  restart  command before the code.
2. Upload here your worksheet or the full text of the code. 

@ThU   in  Maple 2017.3 Windows 32 bit

@asa12 

1. You wrote "is it possible sqrt(3)/6 and 1/2 combine into one term and the rest is one matrix result?". It's impossible.

2. I do not have Maple 12. I did this in Maple 2017.3

Maple by default chooses pretty close colors for graphs, so for greater clarity I suggest setting colors by yourself. scaling  and  size  options are also very useful:

exprs:=[sin(x), cos(x), sqrt(x)*sin(x)];
plot(exprs, x=0..6, color=[red,blue,green], scaling=constrained, legend=exprs, size=[700,400]);

@mrmusic1994  k>0  follows from the physical meaning of the problem, because the resistance force is opposite to the direction of motion.

@tomleslie Thank you for your observation, which allowed me to correctly understand the condition of the problem. At first I understood "right or left" relative to the original coordinate system. In my understanding, this is similar to the movement of a drunken man, who is moving steadily forward, but at the same time he randomly swings to the right or to the left.
I added another option, similar to your one, to my answer.

@Carl Love  Of course, I wrote this purely formally, because OP asked for 3d-animation. I agree that from the physical point of view, 2d-animation is more appropriate:

restart;
u := (x,t)->Sum(4*sin((1/20)*r*Pi*x)*sin((1/2)*r*Pi)*cos((1/20)*r*Pi*t)/(r^2*Pi^2), r = 1 .. 100):
plots:-animate(plot, [u(x,t), x = 0 .. 10, thickness=2, color=blue], t = 0 .. 100, frames = 120);
 

@st0812 

u := (x,t)->Sum(4*sin((1/20)*r*Pi*x)*sin((1/2)*r*Pi)*cos((1/20)*r*Pi*t)/(r^2*Pi^2), r = 1 .. 100):
plots:-animate(plot3d, [u(x, t), x = 0 .. 10, t = 0 .. s, view = [0 .. 10, 0 .. 100, -0.7 .. 0.7], numpoints = 10000], s = 0 .. 100, frames = 60);

It's not a bug. In fact, you solve equation  {y(0) = 1, diff(y(x), x) = sqrt(2-2*ln(y(x)^2))*y(x)} . For x>sqrt(2)/2  the solution also will be  y=exp(1)^(1/2) . Therefore, the solution with numeric option is absolutely correct.

See:

Eq1:=
{diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1], 
y(0) = 1};

sol1 := dsolve(Eq1);
maximize(rhs(sol1), location);

Eq2:=
{diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1], 
y((1/2)*sqrt(2)) = exp(1/2)};
sol2:=dsolve(Eq2);

         

Thus, the one of the symbolic answers for any  x  will be

piecewise(x<=sqrt(2)/2,  y(x) = exp(x*sqrt(2)-x^2), y(x)=exp(1)^(1/2));

Tell me what CAS immediately give this result.

Edit.

@st0812  Probably you have an older version of Maple. Try this option:

restart;
lambda:=unapply(5*Pi*sqrt(m^2/16+n^2/4),m,n):
m:=2*i-1: n:=2*k-1: 
u:=(x,y,t)->0.426050*add(add(1/(m^3*n^3)*cos(lambda(m,n)*t)*sin(m*Pi*x/4)*sin(n*Pi*y/2), i=1..50), k=1..50):
plot3d(u(x,y,0.01), x=0..2, y=0..2, style=patchcontour, axes=BOXED);

On my 32 bit Maple 2017.3  (Windows 10)  there are no problems:

restart;
evalf(Int(x*(1-2*x^(3/10))^(10./3), x=0..1));

                        -0.1197165794 - 0.2074116177*I