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

When `Method -> {"EquationSimplification" -> "Residual"`

applies in `ParametricNDSolveValue`

the 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

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!