Finding NDSolve method details

I have eqs about the NDSolve, I know this code given the solving automatically.

How can I find out what method is used behind the scenes? How can I gauge the reliability level, find how many iterations have been used, the order of method. How can I estimate the error?

I found hints on this site, but I still do not fully understand.

It is impossible to say NDSolve has automatically solution for publishing paper?

I used this code related to my system:

r = 0.431201; β = 2.99 *10^-6; σ = 0.7; δ = 0.57; {m = 0.3, η = 0.1, μ = 0.1, ρ = 0.3};   S = {N1'[t] == r N1[t] (1 - β N1[t]) - η  N1[t] I1[t],      I1'[t] == σ + (ρ  N1[t]  I1[t])/( m + N1[t]) - δ I1[t] - μ  N1[t] I1[t]};  c = {N1[0] == 1, I1[0] == 1.22};  Select[Flatten[   Trace[     NDSolve[{S, c}, {N1, I1}, {t, 0, 30}],      TraceInternal -> True]],    !FreeQ[#, Method | NDSolve`MethodData] &] 

but I don’t understand the output.

Orthogonal projection of line onto plane

I have already read the topic Orthogonal Projection of vector onto plane, but I have a different task. I need to make a notebook in Mathematica where the users would be able to manually add parameters of a line and plane equation, given in forms x=x1+ta1, y=y1+ta2, z=z1+ta3, where (a1,a2,a3) are the coordinates of a vector the line is parallel to and ax+bx+cz+d=0 for a plane, and I need both to return the equation of a projection and graphic representation (plotting the solution).

How could I modify the code in previous topic to correspond to what I am looking for?

Thank you in advance for your help.

Are these expressions not equal? Mathematica output is ambiguous

The following plot indicates that the first expression equals the second. But how can I use Mathematica to show that is true:

Plot[(-(Log[(1 - P)/P]/Log[10])) - (Log[-(P/(-1 + P))]/Log[10]), {P, 0, 1}] 

An attempt to simplify indicates the expressions are not equal:

FullSimplify[(-(Log[(1 - P)/P]/Log[10])) - (Log[-(P/(-1 + P))]/Log[10])] 

That gives the following answer:

(Log[-1 + 1/P] + Log[-(P/(-1 + P))])/Log[10] 

enter image description here

Can I get the “Image Size”, etc. from a NetEncoder in an existing net from the Repository?

I want to get the details about the input used to train a net in the Net Repository, in my case “LeNet Trained on MNIST Data”. I can get its decoder by:

NetExtract[ NetModel["LeNet Trained on MNIST Data"] , "Input" ] 

but I cannot figure out how to get properties like Type and ImageSize from inside it.

Equation Simplification and Implicit Runge Kutta

I want to solve a system of ODEs with trapezoidal method. My system of ODEs is so large that I have to use the DAE solver ("EquationSimplification" -> "Residual"). However, I do not know the syntax for incorporating the options (trapezoidal and DAE solver) in NDSolve. I have tried with the following code, but it failed.

Trapezoidal = {"FixedStep", "EquationSimplification" -> "Residual",     Method -> {"ImplicitRungeKutta",       "Coefficients" -> "ImplicitRungeKuttaLobattoIIIACoefficients",       "DifferenceOrder" -> 1,       "ImplicitSolver" -> {"FixedPoint",         AccuracyGoal -> MachinePrecision,         PrecisionGoal -> MachinePrecision,         "IterationSafetyFactor" -> 1}}};  NDSol = NDSolve[{ODEs, ICs0}, Flatten[{dqdt, q}], {t, ti, tf},      MaxSteps -> Infinity, Method -> Trapezoidal][[1]]; 

Please suggest me the correct syntax.

Rearranging similar terms in an equation

In a piece of code I am writing, I run into an equation of the form

eqn = Subscript[x, 3] + Subscript[x, 5] + Subscript[r, 0] + 2 Subscript[\[zeta], 3] + Subscript[\[zeta], 9] == 2 Subscript[x, 4] + Subscript[\[zeta], 6] + Subscript[\[zeta], 7]

In more readable $ \LaTeX$ format, this looks like

$ $ x_3 + x_5 + r_0 + 2 \zeta_3 + \zeta_9 = 2 x_4 + \zeta_6 + \zeta_7$ $

Given such an equation with Subscript[x, __] as the variable of interest, I would like to rewrite it with all $ x_i$ ‘s on the left hand side and everything else on the right-hand side, i.e.

$ $ x_3 – 2 x_4 + x_5 = -r_0 – 2\zeta_3 + \zeta_6 + \zeta_7 – \zeta_9$ $

(I am not interested in the TeXForm of course. That is trivial to extract if needed.)

Now given such an equation (eqn) I can extract the left and right hand sides as

lhs = eqn[[1]]; rhs = eqn[[2]];

Then the variables on each side are

vLHS = Variables[lhs]; vRHS = Variables[rhs];

But how does one rearrange the equations like so?