diffeq | Basic Ordinary Differential Equation solvers

I am trying to solve a Differential Equation with 4th Order Runge - Kutta method in Python3. The function is deliberately made so that the solution goes to infinity.

My issue is that I have been asked to plot the iterative solutions, but cant seem to understand how to handle the infinity, when storing the iterator and the respective solution values in a list inside the for loop, and when I call that list outside the loop. Below is the entire code and result pane with the error message.

I am making an separable differential equation solver. In order to make an expression that separated by x and y variables I have to divide expression on the right by every variable that belong to s such as sin(y), e**y, y**2, ... I am using Sympy

I am learning sympy and first order linear difference equations.

The solution is y(n) = n^{2}*.25 + n*.625 + 0.28125*(1-(-3)^{n}) for the equation y(n) = x(n)-3y(n-1) with initial conditions y(-1)=0 and x(n) = n^{2}+n.

I am stuck at solving, this is what I have:

I'm trying to simulate a reflecting boundary. Based on the suggestions found here: Stochastic differential equation with callback in Julia I tried

I'm trying to write an integrator which uses long doubles for very high precision. I know that my system architecture has long double support, but for some reason, the precision of my integrator maxes out at 16 significant digits. Here's some code which recreates what I'm seeing. The integrator for this example was adapted from this source. In this test case, I am using it to calculate Euler's number (I apologize for the length of the code block but I can't recreate the behavior any other way):

Let's suppose the example 1 of the bouncing ball with multiple walls in the page:

https://diffeq.sciml.ai/stable/features/callback_functions/

And consider the condition:

I want to implement the adjoint sensitivity analysis in python, in order to determine the gradient of my objective function with respect to some parameters. In specific the objective function depends on the solution of a differential equation which in turn depends on said parameters which I am looking to find the optimum of.

To perform this there are numerous good packages both in Julia (see here), as well as CVODES from SUNDIALS, however the latter which does apparently have a wrapper made for python, does not include sensitivity analysis capabilities according to this link. Furthermore, I have looked into SALib for sensitivity analysis, but as far as I understand this refers to some other type of 'sensitivity analysis' and therefore adjoint or even forward sensitivity analysis is not included (correct me if I am wrong on this one).

Thus my question is, does a version of CVODES exist in python with sensitivity analysis capabilities, or is there there any other package where one can use in order to perform adjoint sensitivity analys?

I am trying to get the differential equation y'=sin(x) however my differential equation will not run any further as I get the error "can't convert expression to float". If I use numpy with np.sin(x), I get another error "loop of ufunc does not support argument 0 of type Symbol which has no callable sin method". Here is the code:

I want to solve the matrix-form time-dependent Schrodinger equation on 3d lattice with DifferentialEquations.jl,
i.e., (∂/∂t)ψ = -iHψ ,where ψ is a vector and H is a (time-independent) matrix.
I tried to write the code like this.