polynomials – Solution of the Maclaurin series to the Legendre equation and general expression of the coefficients

Consider a Maclaurin series solution
$$ y = (1-x ^ 2) y '' -2xy '+ α (α + 1) y = 0, -1 <x <1. $$

CA watch $$ a_2 = frac {-α (α + 1)} {6} a_0 $$ $$ a_3 = frac {- (α – 1) (α + 2)} {6} a_1 $$

and for all $ n≥2 $,
$$ a_ {n + 2} = frac {n (n + 1) – α (α + 1)} {(n + 2) (n + 1)} a_n = frac {(n – α) (n + α + 1)} {(n + 2) (n + 1)} a_n. $$

Deduce this, if $ α = k ∈ {0,1,2,3, … }, $ then
$$ a_ {k + 2} = a_ {k + 4} = a_ {k + 6} = … = 0. $$
Therefore, write a polynomial solution of the Legendre equation in the cases $ α = $ 0,1,2,3,4.
For $ α = $ 3, write the first 4 terms of the solution of the Legendre equation in the other series and try to find a general expression for the coefficients of this series.

calculation and analysis – The resolution command does not solve this equation!

I've tried to solve the following equation with Mathematica:

$ left (2-x ^ 2 right) left (n left (x ^ 4-x ^ 2 + 1 right) – pi left (x ^ 2-2 right) right) sinh ( pi x) cosh (nx) + sinh (nx) left ( left (2-x ^ 2 right) left ( pi left (x ^ 4-x ^ 2 + 1 right) -n left (x ^ 2-2 right) right) cosh ( pi x) -x left (x ^ 4 + 2 x ^ 2 right) sinh ( pi x) right) = $ 0

but the answer is:
"This system can not be solved with the methods available to solve."

I've also tried Maple, the result has been a long relationship in terms of RootOf.
How can I get an explicit solution for $ x $ in terms of $ n $?

Initial value problem in an integral equation of Volterra

How can I transform

$$ frac {d ^ 2u} {dt ^ 2} = f (t, u), u (0) = u_0, u (u) = u_1 $$
in an integral equation of Volterra?

Error trying to solve a linear differential equation

I'm trying to solve a linear differential equation for changing the scale factor term, $ delta a (t) $when mass terms are introduced into the equations of motion for general relativity. The first equation I must solve is:

$$
frac {d} {dt} delta a (t) = frac {8 pi G (a (t)) ^ 2} {3 frac {d} {dt} a (t)} (3m ^ 2 (a (t)) ^ 2 mp frac {1} {2} delta rho (t)) + frac {2 frac {d} {dt} a (t) delta a (t)} {a}
$$

This equation is of the form $ frac {d} {dt} delta a (t) + f (t) delta a (t) = g (t) $which is a linear differential equation.

However, when I plug this equation into Mathematica using the DSolve function, I get an error of the degree of

Equation or list of equations expected instead of True in the first argument

My code is as follows:

DSolve({(Delta)a'(t) - (2*a'(t)*(Delta)a(t))/a(t) == (
    8*(Pi)*G*(a^2)(t))/(
    3*a'(t)) (3*m^2*(a^2)(t) - 1/2 (Delta)(Rho)(t)), (Delta)a(0) ==
    0, a(0) == 1}, (Delta)a(t), t)

where I introduced the conditions $ delta a (0) = 0 $ and $ a (0) = $ 1.

What am I doing wrong here, is it simply impossible to solve numerically?

Thank you!

ap.analysis of pdes – Green's function for 3D relativistic heat equation

On the Wikipedia page, it is indicated that Green's function for relativistic 3D thermal conduction (with $ c = $ 1)

$$ ( partial_t ^ 2 + 2 gamma partial_t – Delta_ {3D}) u (t, x) = delta (t, x) = delta (t) delta (x) $$

is given by
$$ u (t, x) = frac {e ^ {- gamma t}} {20 pi} bigg ( big (8 – 3rd ^ {- gamma t} + 2 gamma t + 4 gamma ^ 2t ^ 2 big) frac { delta (t- | x |)} {| x | ^ 2} + gamma ^ 2 Theta (t – | x |) big ( frac {I_1 big ( gamma sqrt {t ^ 2 – | x | ^ 2} big)} { sqrt {t ^ 2 – | x | ^ 2}} + t frac {I_2 big ( gamma sqrt { t ^ 2 – | x | ^ 2} big)} {t ^ 2 – | x | ^ 2} big) bigg) $$

Or $ I_1 $ and $ I_2 $ are modified Bessel functions of the 1st and 2nd types. In addition, $ Theta $ is the Heaviside step function. I am looking for a derivation of this or a reference to one.

Gentle attempt:
Taking the Fourier transform with regard to $ x $ we obtain
$$ ( partial_t ^ 2 + 2 gamma partial_t + (4 pi ^ 2 | xi | ^ 2)) hat {u} (t, xi) = delta (t) $$

On the same Wikipedia page, there is a 1D damped harmonic oscillator and they claim that Green's function is given by
$$ hat {u} (t, xi) = Theta (t) e ^ {- gamma t} frac { sin big (t sqrt {4 pi ^ 2 | xi | ^ 2 – gamma ^ 2} big)} { sqrt {4 pi ^ 2 | xi | ^ 2 – gamma ^ 2}} $$
So I should wait for that

$$ u (t, x) = int e ^ {2 pi i xi cdot x} Theta (t) e ^ {- gamma t} frac { sin big (t sqrt {4 pi ^ 2 | xi | ^ 2 – gamma ^ 2} big)} { sqrt {4 pi ^ 2 | xi | ^ 2 – gamma ^ 2}} d xi $$

But I'm not sure how we would get the formula above or if it's even the right approach.

divide and conquer – recursive equation of this code

power (x, n) {
if n == 0
back 1
if n is even
return power (x, n / 2) * power (x, n / 2)
if n is odd
return power (x, n / 2) * power (x, n / 2) * x

what will be the receipt for this code and how will you get a closed form using the substitution method

General solution of the differential equation: y = (y + x + 1) / (2y + 2x-1)

I am relatively new to this lesson and neither I nor my class know it well. We received a simple and orderly equation to find a solution via the Internet.
$$ y = (y + x + 1) / (2y + 2x-1) $$
Thank you for your time.

numeric value – Laplace equation DirichletCondition ignored

My goal is to find the force between two identical parallel disks with a voltage difference.

To solve this problem, I tried to use Laplace 's equation.
And below is my code.
where R is the radius of the disks, d is the distance between two plates, V0 is the voltage difference between them.

R = 0.1;
e0 = 8.854187817*^-12;
d = 0.48*10^-2;
V0 = 4000;

regionCyl = 
 DiscretizeRegion[
  ImplicitRegion[Sqrt[x^2 + y^2 + z^2] <= 0.5, {x, y, z}], 
  PrecisionGoal -> 6]
laplacian = Laplacian[V[x, y, z], {x, y, z}];
boundaryCondition = {DirichletCondition[V[x, y, z] == V0/2, 
    0 <= x^2 + y^2 <= R^2 && z == d/2], 
   DirichletCondition[V[x, y, z] == V0/2, 
    0 <= x^2 + y^2 <= R^2 && z == -d/2], 
   DirichletCondition[V[x, y, z] == 0, x^2 + y^2 + z^2 == 10]};
sol = NDSolveValue[{laplacian == 0, boundaryCondition}, 
   V, {x, y, z} [Element] regionCyl];
electricField[x_, y_, z_] = -Grad[sol[x, y, z], {x, y, z}];

But with this code I have a message
enter the description of the image here

Thank you for reading.

Equation Resolution – Why is FindRoot so slow for this problem? Det[M[x]]== 0

I'm trying to use Mathematica to find the numerical solution to an equation Det (M (x)) == 0, where M (x) is a matrix function of x, defined below:

r = 10; (Omega) = E^((Pi) I/r); 
M(x_) = Table(
  D(Cos((Omega)^j x), {x, i}), {i, 0, r - 1}, {j, 0, r - 1}); 
G(x_) := Det(N(M(x))); M(x) // MatrixForm

I'm waiting to see the solution to G (x) == 0 for big r up to a few hundred, but when I call the FindRoot function, it's already very slow even for r = 10 (does not end in a few seconds):

FindRoot(G(x) == 0, {x, (r + 1)/4 (Pi)})

But the plot of G (x) is extremely fast:

Plot(Norm(G(x)), {x, (r + 0.6)/4 (Pi), (r + 0.7)/4 (Pi)})

And in fact, I tried to manually use the bisection method and trace again and again (decreasing the interval by a factor of 2 every time), which gives me a very precise solution in just one minute, much faster than FindRoot.

So, why is FindRoot so slow in this case?

Equation Resolution – Solve works with symbols, not with numbers

I do not understand why, in the code below, Solve works with symbols, but substituting numeric (rational) values ​​for symbolic parameters makes it insoluble:

Clear(mo, RT, n)

dG = mo + 2 RT Log(n/(1 + n)) - RT Log(-1 + 2/(1 + n))
Solve(dG == 0, n)

vals = {mo -> 79900, RT -> 6197399567/2500000};
dGvals = dG /. vals
Solve(dGvals == 0, n)

Mathematica 11.2.0.0 on Mac OS X Yosemite