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- Homework Statement
- For the DE ##\frac{dy}{dx} = y^2-x,## the long term behaviour of the Euler approximations with step sizes ##h = 0.5, 0.25, 0.125## with initial condition ##(-1,0)## all tend to ##\infty##. However, the long term behaviour of the actual solution with the same initial condition actually tends to ##-\infty##.

What would I call this failure——failure of self-consistency, failure of convergence, failure of structural stability, or failure of stability?

- Relevant Equations
- N/A

I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not sure which failure this falls under...