Given the ODE $\frac{dy}{dt} = f(t,y)$ and the function $f(y) = -y^3$, with the initial condition $y(0)=1$, I want to use the backward Euler Method with $h = \frac{1}{2}$, combined with the Newton-Rapson method to approximate $y$ at $t = 2$.
This ODE is seperable and an exact solution can be found, which I found to be $y = \sqrt{\frac{1}{2t+1}}.$ Therefore at $t = 2$ the exact solution would be $\sqrt{\frac{1}{5}}$.
However, I want to know how to use these two numerical methods to approximate a solution. So far, I used the Backward Euler Method as follows:
$$y_{n+1} \approx y_n + hf(t_{n+1}, y_{n+1})$$ $$y_{n+1} \approx 1 - \frac{1}{2}y_{n+1}^3$$
This forms the cubic equation $y_{n+1}^3 + 2y_{n+1} - 2 \approx 0$ which can be solved via Newton-Raphson, and this gives me a value of $y_{n+1} = 0.7709$. But where can I go from here? Have I made a mistake? Any guidance would be much appreciated.