You have two errors:

1. Pi must be spelled with capital P and lowercase i,
2. polarplot should be plots:-polarplot.

It can be done with geom3d:-intersection. In particular, if the result of this command is not NULL, then the input objects intersect.

It can be done in two steps:

convert(s, trig);
combine(%);

f:= 3+2*x+y+2*x^2+2*x*y+3*y^2 = 0:
d1:= implicitdiff(f, y, x);

d2:= implicitdiff(f, y, x$2);

This can be simplified substantially using the original expression and simplify with side relations:

simplify(d2, {f});

diff(f(x,y)=exp(2*x+cos(y)), y$3, x$5);

Edit menu => Split or Join => Split Execution Group (F3)

Simply apply command factor to the result returned by intsolve:

intEq:= 2*cosh(x)-sinh(x)-(2-x)= 1 + int( (2-x+t)* phi(t),t=0..x):
Sol:= intsolve(intEq,phi(x)):
factor(Sol);

Try adding the following option to the plot commands:

xtickmarks= [38320= "11/29", 38330= "12/09", 38340= "12/19", 38350="12/29"]

If you have a lot of such plots to do, it may be worthwhile to write a procedure that translates dates from this "Excel format" to a standard format. This can be done with Finance:-AdvanceDate using any known match (such as 38320 = "11/29/2004") as a reference point. The date can be formatted into a string with Finance:-FormatDate.

A search at the Maple Applications Center turned up one hit: here .

Use Zoom factor from the View menu.

Eigenvalues is in the module (package) LinearAlgebra. So, to use it, you need to either load the package with with(LinearAlgebra) or use the module name as a prefix. I recommend the latter. So, change Eigenvectors to LinearAlgebra:-Eigenvectors.

Assign the solution returned by solve to a variable. Like this

Sol:= solve( ...the rest of your solve command);

Then do

f:= unapply(eval(f(x), Sol), x);
g:= unapply(eval(g(x), Sol), x);

I don't believe your plot. How did you make it? I get that f(2) is about 10^7.

High-degree polynomials are notoriously unstable for numeric evaluation. You should use numeric integration. (Although, as noted by nm, the regular integrator seems to give the correct value also.) The numeric integrator will not give an answer if it cannot achieve the requested accuracy. You need to use the digits option (or the epsilon option) to make the accuracy request less than your value of Digits, which is 10 by default.

evalf(Int(f(x), x= 0..2, digits= 6));

I just picked digits=6 arbitrarily. You can make it larger, as long as Digits is somewhat larger than digits.

You must spell infinity with a lowercase i.

Numeric approximation says that your integral diverges to infinity:

Integrate(BesselJ(0,x)^2, x=0..infinity);

value(%);

evalf(%);

Two issues before answering your question:

1. You need to use `*` to represent all multiplications.
2. Once your polynomial is corrected with respect to (wrt) #1, it is already collected wrt the desired subpolynomial.

To collect wrt a subpolynomial, you need to first substitute a single name for the subpolynomial:

g:= -2-k[1]*(lambda*alpha[2]*k[2]+alpha[1]*(-2-2*k[2]+k[2]*lambda^2))+k[1]*(lambda*alpha[2]*k[2]+alpha[1]*(-2-2*k[2]+k[2]*lambda^2))*lambda*(lambda*alpha[2]*k[2]+alpha[1]*(-2-2*k[2]+k[2]*lambda^2))^2:

h:= lambda*alpha[2]*k[2]+alpha[1]*(-2-2*k[2]+k[2]*lambda^2):

subs(h= E, g);

collect(%, E);