According to Wikipedia, "d'Alembert's Test" is called "ratio test" in English.
Try it on series
sum(1/(2^n+3^n),n=1..infinity);

> f := x -> a*x^3+b*x^2+c*x+d;
f := proc (x) options operator, arrow; a*x^3+b*x^2+c*x+d end proc
> locmax_conditions := {f(-1)=-2,D(f)(-1)=0};
locmax_conditions := {3*a-2*b+c = 0, -a+b-c+d = -2}
> locmin_conditions := {f(4)=4, D(f)(4)=0};
locmin_conditions := {48*a+8*b+c = 0, 64*a+16*b+4*c+d = 4}
> solve(locmax_conditions union locmin_conditions);
{a = -12/125, b = 54/125, c = 144/125, d = -172/125}
> subs(%,f(x));
-(12/125)*x^3+(54/125)*x^2+(144/125)*x-172/125

And probably the professor wanted *him* to program it, rather than using some pre-programmed "lambda" function.
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G A Edgar

> solucion[1] := {x=2,y=3};
> solucion[2] := {x=3,y=4};
> solucion[3] := {x=2,y=0};
solucion[1] := {x = 2, y = 3}
solucion[2] := {x = 3, y = 4}
solucion[3] := {x = 2, y = 0}
> for i from 1 to 3 do
> x[i] := subs(solucion[i],x);
> y[i] := subs(solucion[i],y);
> end do;
x[1] := 2
y[1] := 3
x[2] := 3
y[2] := 4
x[3] := 2
y[3] := 0
> x[2];
3

]]How do I display the solutions it has when it says some may have been lost?
I believe it first displays all the solutions it has, then it says "some solutions may have been lost"
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G A Edgar

Re(int(1/x,x));
ln(|x|)
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G A Edgar

plot([x^2/20,sin(x),2-x], x=0..5, color=[red,black,green]);
If you live in Canada (or even if you don't) you can write colour for color.
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G A Edgar

# try everything up to 1000, if no result increase the bound
for N from 1 to 1000
if irem(N,21)=19 then
......
end if;
end for;
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G A Edgar

Maybe it would be better to use the regular int, rather than the inert substitute Int ??
It looks like just a polynomial in t, so after integration it should be much easier to work with,
even if it has giant coefficients.
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G A Edgar

N. G. De Bruijn, "The Difference-Differential Equation F'(x) = e^(alpha*x+beta)*F(x-1)."
Indagationes Math. 15 (1953) 449--464.
If you write F(x) = e^(-x)*f(x), your equation becomes one of his.
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G A Edgar

Must be those pirates operating off Somalia.
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G A Edgar

It is hard to tell what you mean. Maybe give an example.
Perhaps you need to increase the printlevel ... are your prints inside a loop or procedure?
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G A Edgar

**f := (x,t)->t-x^3;**
f := proc (x, t) options operator, arrow; t-x^3 end proc
**f(x,t)=0;**
t-x^3 = 0
**fsolve(f(x,t)=0,x);**
Error, (in fsolve) t is in the equation, and is not solved for
**g := unapply('fsolve(f(x,t)=0,x)',t);**
g := proc (t) options operator, arrow; fsolve(f(x, t) = 0, x) end proc
**g(0.2);**
.5848035476
**evalf(Int(g,0..1));**
.7500000000

Certain definite integrals of this kind are known...
**int(cos(a*sin(t)),t=0..Pi/2);**
(1/2)*Pi*BesselJ(0, a)
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G A Edgar

You want a closed-form solution?
Why do you think there is one?
Even this simplified equation is not solved by Maple:
> de1 := diff(y(x),x) = 1/(y(x)-sin(x));
de1 := diff(y(x), x) = 1/(y(x)-sin(x))
> dsolve(de1,y(x));
>
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G A Edgar