Differential equations – Some problems with DEingesystem

I would like to solve (get its values ​​/ eigenvectors) the Sturm-Liouville problem, for the following differential operator: $ L = partial_ {r} partial_ {r} psi (r) $. Also, I would like to impose the following boundary conditions:$ B1 = psi (0) = 1, B2 = psi (2) = 0. $

I have tried several different things and the only one that seems to work is the following:

L = D[[Psi][r], r, r];

[Lambda] = 2;

B1 = [Psi][0]    == 1;

B2 = [Psi][2]    == 0;

{ev, ef} = DEIGENSISTEMA[L[L[L[L[Psi][r], {r, 0, 2}, 2]Plot[{Evaluate[ef], function}, {r, 0, 5}]

It is clear that conditions were not solved in this example. I am very aware that, within the documentation, it is written about how to use the condition of Dirichilet, as an example, but not for general conditions of limit / initials.

With this, my questions are the following:

1) Why only accept second order operators?

2) What conditions do you use when I do not specify it (as in my example)?

3) Adjusting around with the written part $ {r, 0.2 }In the DEigensystem function, it seems that conditions are changing, but I do not see how. What exactly is this term doing?

4) And the most important question for my problem is: how can I establish two arbitrary conditions (like $ B1 $ Y $ B2 $ given in my example)?

By parts with differential equations – Mathematica Stack Exchange

I have the following differential system. The calibration that I use is the following

paramFinal = {rho -> 0.025, R -> 0.25, alpha -> 0, k -> -0.55, g -> 1.2, C0 -> 3, C1 -> 0.1, sbar -> 10, [Eta] -> 8.5, hbar -> 0.5, Hbar -> 23.5};

The differential system is

dec = W & # 39;

des = H & # 39;

With NDSolve, I can solve the problem without any problem, but I would like to restrict the variable $ H

pdec = a pieces[{W'[{W'[{W'[{W'

pdes = a pieces[{H'[{H'[{H'[{H'

nsba3aa = NDSolve[{pdec /. paramFinal, pdes /. paramFinal, W[0] == 0.275, H[0] == 25, {W

I have the following error message and I can not find out where the problem is

By pairs :: pairs: The first argument {(W ^[Prime])
R) / W
H
pairs >>

Differential equations. How could I know the method adopted in NDSolve?

In NDSolve, different methods can lead to different results. For my codes

Clear["Global`*"]
α = 110; β = 55; δ = 1; μ1 = 18; μ2 = 42; μ = μ2 / μ1;
ηb = 10; deltap = .18; p0 = 0.03; T = 1.0;
f0 = T / (2 [Pi]); n = 1; fn = n * f0; reference point = 4; tlength = 1000;
w[λ_, ξ_] : = (- ((μ1 * α) / 2) Log[ 1 - (λ^(-4) + 2*λ^2 - 3)/α] 
           - (μ2 * β) / 2 Log[ 1 - (λ^-4*ξ^4 + 2 λ^2*ξ^-2 - 3)/β]) / μ1
dw[μ_, ξ_] = D[w[λ, ξ], λ];
in[λ_, ξ_, x_] = (1 + (λ ^ 3 - 1) (x ^ 3 - 1) ^ - 1 (ξ ^ 3 - 1)) ^ (1/3);
F[λ_, ξ_, x_] = dw[λ, ξin[λ, ξ, x]]/ (1 - λ ^ 3);
dine[x_] : = ((δ + x ^ 3) / (1 + δ)) ^ (1/3)
Obtain["NumericalDifferentialEquationAnalysis`"];
np = 11; points = weights = table[Null, {np}];
intf[x0_, ξ0_] : =
To block[{y = x0, ξ1 = ξ0}, 
  Do[points[[i]]=
Pesos[np, y, sup[y]][[i, 1]], {i, 1, np}];
Do[weights[[i]]=
Pesos[np, y, sup[y]][[i, 2]], {i, 1, np}];
int = Sum[(f[λ, ξ1, y] /. λ -> points[[i]]) *
pesos[[i]], {i, 1, np}]; In t]eq1: = x & # 39; & # 39;
2 x & # 39;
X
3 (1 + δ / x
deltap - p0 * Sin[two[two[2[2[Pi]* fn * t]) /
X
eq2: = (3 ηb * (1 - (x
3) / β)) ξ & # 39;
     t]* (μ (x
t0 = Time used[];
pfun = ParametricNDSolveValue[{eq1, eq2, ξ[0] == p, x & # 39;[0] == 0,
X[0] == p}, {x
Method -> {"EquationSimplification" -> "Residual"} *)];
p3 = pfun[inipoint];
Time used[] - t0
ParametricPlot[Evaluate@p3[[1 ;; 2]], {t, 0.9 * tlength, tlength},
PlotPoints -> 60, PlotStyle -> Blue, PlotRange -> {{3, 8.}, All},
AspectRatio -> GoldenRatio ^ -1, AxesOrigin -> {0, 0}, Frame -> True,
FrameStyle -> Directive[Black, 12]]

The result is
Figure 1

When Method -> {"EquationSimplification" -> "Residual" applies in ParametricNDSolveValuethe code is

pfun = ParametricNDSolveValue[{eq1, eq2, ξ[0] == p, x & # 39;[0] == 0,
X[0] == p}, {x
Method -> {"EquationSimplification" -> "Residual"}];
p3 = pfun[inipoint];
Time used[] - t0
ParametricPlot[Evaluate@p3[[1 ;; 2]], {t, 0.9 * tlength, tlength},
PlotPoints -> 60, PlotStyle -> Blue, PlotRange -> {{3, 8.}, All},
AspectRatio -> GoldenRatio ^ -1, AxesOrigin -> {0, 0}, Frame -> True,
FrameStyle -> Directive[Black, 12]]

the result is
<img src = "https://i.stack.imgur.com/IIAs0.jpg

We can find some differences for the two diagrams when a specific method is applied in ParametricNDSolveValue.

And if the equations are written as differential equations of an order as below

eq1: = x & # 39;
eq2: = y & # 39;
X
3 (1 + δ / x
deltap - p0 * Sin[two[two[2[2[Pi]* fn * t]) /
X
eq3: = ξ & # 39;
X
2 x

pfun = ParametricNDSolveValue[{{eq1eq2eq3}{x[{{eq1eq2eq3}{x[{{eq1eq2eq3}{x[{{eq1eq2eq3}{x[0] == inipoint,
Y[0] == 0, ξ[0] == inipoint}}, {x
length}, {p},
Method -> {"EquationSimplification" -> "Residual"}];
p3 = pfun[inipoint];
ParametricPlot[Evaluate@p3[[1 ;; 2]], {t, 0.9 * tlength, tlength},
PlotPoints -> 60, PlotStyle -> Blue, PlotRange -> {{3, 8.}, All},
AspectRatio -> GoldenRatio ^ -1, AxesOrigin -> {0, 0}, Frame -> True,
FrameStyle -> Directive[Black, 12]]

The result is
Fig. 3

Another different result is given.
Questions:
1. For the same equations, I just specified the Method or change the shape of the equations, why are the results different and which is better?
2. Is it possible to know the specific method or the strategy adopted in functions such as NDSolve Y ParametricNDSolveValue?
Any idea would be greatly appreciated!

Differential geometry: How to prove that the two topologies in an embedded sub-matrix are actually the same?

Leave $ M, N $ be two smooth collectors and $ F: M rightarrow N $ Be a soft immersion injective. We know that with subspace topology, yes $ F: M rightarrow F (M) $ it's a homeomorphism, then $ F (M) $it is an embedded submanifold of $ N $.

However, we can also equip $ F (M) $ with such topology:
$$ A subset F (M) text {is open if and only if} F ^ {- 1} (A) subset M text {is open} $$

My teacher said that the two topologies are actually the same when the case is insufficient. But it still confuses me why? How can I prove this fact?

Differential equations – Limit value problem for non-linear EDO

I'm trying to solve the following limit value problem using math v12:

$$
(1-y) (1+ (1-y) yq (y) (2 + yq (y))) q ^ { prime prime} (y) -q (y) ^ 2 (2-5 y + (2-3 y) yq (y)) – 2 (1- (1-y) q (y) (1-5 y + y (1-3 y) q (y))) q ^ prime (y ) + (1-y) ^ 2 y (1 + yq (y)) q ^ prime (y) ^ 2 = 0
$$

with boundary conditions

begin {align}
& q ^ prime (0) -q (0) – frac { epsilon ^ 2} {8} = 0 ,,
\
& q ^ prime (1) – frac {1} {2} q (1) ^ 2 (3 + q (1)) = 0 ,.
end {align}

However, I still receive the following error (for all values ​​of $ epsilon $ I have tried):

NDSolve :: ndsz: In y == 0.999999999999854857`, the step size is effectively zero; The singularity or rigidity of the system is suspected.

Any help would be greatly appreciated.

Ordinary differential equations – Given the curve y = x3 -3×2- 18x. Find the critical point, maximum and minimum and draw the curve.

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Differential equations: Vasicek model and sports interest rate parameterized by the reversal rate.

When solving an SDE, I want to derive the analytic results from the mean and variance of the extended Vasicek model process.

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while gamma is the rate of reversion and s = eta / gamma the average short rate

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while gamma is the rate of reversion and s = eta / gamma the average short rate

How can I configure Xt = rt – s and solve by integrating on both sides of the SDE with the help of the integrating factor e ^ yt and in a second step derive the mean and variance?

Differential equations – How do I solve my PDE system?

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differential: how does the integral constraint for a PDE on a volume equal to zero guarantee the uniqueness?

I am studying the optimization of the partial differential equation, and this is the problem of the elliptical model:

(one) $ – nabla space cdot (e ^ u nabla y_i) = q_i space space space x en Omega $

(two) $ nabla and_i space cdot space n = 0 $

(3) $ int_ Omega y_i space d Omega = 0 space space space i = 1, …, n_s $

Where $ Omega subset mathbb {R} ^ 3 $. My biggest problem is equation 3, which is said to insure the singularity. I know that $ n $ is the normal vector in the limit $ partial Omega $ but I can only guess $ n_s $ It's the number of normal vectors, but I'm not totally sure. I know that this means that the integral is above all the volume, but it's also equal to zero, so that does not make every y a point? Also, how does the $ i $ subindex factor in this?

Differential equations – DSolve of second order DE results n InverseFunction

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