## 16048 Reputation

12 years, 197 days

## No solutions for x<=0...

Your function y(x)=-2*x^2+C1*ln(x)  is the solution of a differential equation

x*ln(x)*diff(y(x),x)-y(x)=2*x^2-4*x^2*ln(x)  which has no solutions for x<=0, since the domain of the function ln (x) only positive numbers.

## Write your last line as follows and get ...

Your decision will be correct if the line MN does not intersect the sphere S. Otherwise, the answer is 0. If the point M(1, 2, -1) then at the end of your code, add the command allvalues(%)

## The idea of ​​the solution...

Let O be a center of the sphere.

1) Find the equation of the plane passing through the points M, N and O.

2) Find the equation of the line of intersection of this plane and the sphere.

3) The points for which the distance from the sphere to the line MN the maximum and minimum are on the line.

## n different real roots...

If n is known, then your equation is reduced to finding the roots of a polynomial of degree n. It is easy to prove that, for different  a_i, this polynomial will have n different real roots. But finding them in a closed form for large n seems impossible. See the example for n = 5:

restart:

f:=unapply(sum(1/(x+a[i]),i=1..n), (x,n));

n:=5;

a:=[seq(k,k=1..n)];

f(x,n);

solve(f(x,n)=1);

allvalues([%]);

evalf(%);

plot([f(x,n),1],x=-18..8,-3..5, color=[blue,red], thickness=2, discont=true);

## Another variant...

Try this variant:

restart;

k := 0; B[1] := 0.001/365; B[2] :=0.002/365; B[3] :=0.005/365:

ode := diff(U(t), t) = -(A[1]+A[2]*U(t))*U(t);

ic[0] := U(365*k) = 1000;

sol[0] := dsolve({ic[0], subs(A[1] = B[1], A[2] = B[2], ode)}, U(t), numeric);

sigma := .5;

for k to 4 do

if k = 3 then A[1] := B[3]:

ode1 := diff(U(t), t) = -(A[1]+A[2]*U(t))*U(t):  end if;  tk := 365*k;

V := rhs(sol[k-1](tk)[2]);  ic[k] := U(tk) = sigma*V;

if k=3 then sol[k] := dsolve({ic[k], subs(A[2] = B[2], ode1)}, U(t), numeric); else

sol[k] := dsolve({ic[k], subs(A[1] = B[1], A[2] = B[2], ode)}, U(t), numeric); fi;

end do;

## Maple already has such a procedure: cei...

Maple already has such a procedure:

ceil(3.5);

4

## The idea of ​​the solution...

If i understand your problem, you're trying to experiment to find a straight line passing through the point (4, 4/3) and to be tangent to semicircle  y=-sqrt(1-x^2)+3 .  This problem can be solved exactly, as follows:

1) At any point of the semicircle find the tangent line to this semicircle. You get the equation of a straight line, depending on the parameter x that specifies the position of the point of tangency.

2) Using the condition that the line passes through the point  (4, 4/3) , find the point of tangency, and then your desired angle.

I got the angle phi=arctan((20-4*sqrt(10))/45)=0.1619229414

## Slip of the pen...

Sorry, this is slip of the pen. In the code, replace everywhere 2^(u*v) by 2*u*v.

## Solution...

Make it as follows:

A:= `2*(a+b)`:

B:= `2*a+2*b`:

evalb(A = B);

false

## Solution...

Blue color- x = constgreen colory = const:

plots[implicitplot]([seq(u^2-v^2=C, C=-10..10), u^2-v^2=-1/3, u^2-v^2=1/3, seq(2^u*v=C, C=-10..10), 2^u*v=-1/3, 2^u*v=1/3], u=-10..10, v=-10..10, color=[blue\$23, green\$23],numpoints=10000, view=[-1..3, -3..3], thickness=3,scaling=constrained);

## Statistics[NonlinearFit]...

If the parameters C1 and C2 are unknown, then use the command  Statistics[NonlinearFit]

## Why is m not zero? It is natural to...

Why is m not zero? It is natural to assume that m is any none-negative number:

int((x-m^2)/(x+1),x=0..m^2)  assuming m >= 0;

-ln(m^2+1)+m^2-ln(m^2+1)*m^2

## A more general approach...

It can be generalized to the continuous case (integration), but the previous two approaches can't.

int( int(x*y, y=1..piecewise(8-x>6, 6, 8-x)), x=1..6);

1627/12

Compare:

int( int(x*y, y=1..min(6, 8-x)), x=1..6);

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