Linearization such system of ODEs

Question

I have the following systems of ODEs

$$\begin{aligned} \dot{x} &= -x^2 + \frac{1}{y+1} + 1 \\ \dot{y} &= 1 \end{aligned}$$

and I am trying to linearize the system in the classical way using the Jacobian. In Mathematica,

asys = AffineStateSpaceModel[{x'[t] == -x[t]^2 + 1/(y[t] + 1) + u[t], 
    y'[t] == 1}, {{x[t]}, {y[t]}}, {u[t]}, {x[t]}, t] ж

or2 = OutputResponse[asys, 1, {t, 0, 20}];

Plot[{or2}, {t, 0, 20}, PlotRange -> Full];

eq = N[Normal[Solve[{-x[t] + 1/(y[t] + 1) + 1, 1} == {0, 0}, {x[t], y[s]}, Reals]]];

I am stuck at the last stage where the equilibrium points of the vector field are sought. Stuck because the right side of second equation is $1$ and it can't be $0$ anywhere. How to get around this obstacle?

I would be grateful for help.

Answer 1

$$\begin{cases} \frac{dx}{dt} = -x^2 + \frac{1}{y+1} + 1 \\ \frac{dy}{dt} = 1 \quad\implies\quad y(t)=t+c_1 \end{cases}$$ $c_1$ is an arbitrary constant insofar no initial condition is given. $$\frac{dx}{dt} = -x^2 + \frac{1}{t+c_1+1} + 1 $$ This is a Ricatti ODE which is analytically solvable. The solutions involve some confluent hypergeometric functions and/or associated Laguerre polynomials.

https://www.wolframalpha.com/input/?i=x%27%28t%29%3D-x%28t%29%5E2%2B1%2F%28t%2Bc%2B1%29%2B1

These complicated solutions might be simplified to functions of lower level depending on the kind of initial conditions.