Question: Given the differential equation and its initial condition as $$dy/dx =-20xy^2 \quad\text{and} \quad y(1)=1$$ respectively, determine, under what conditions each of:
(i) Euler’s method;
(ii) the classical fourth-order Runge-Kutta method;
is stable over the interval $1 \leq x \leq 2$, given that for this problem the maximum value of $xy$ occurs when $x = 1$.
I do not understand what this question is asking for. Any hint is appreciated.
The first observation is that if $y$ at some time during the numerical method takes a negative value, the exact solution will diverge towards $−∞$, and the numerical solution will follow. So what is observed is at least a move further away from the $x$-axis inside the integration interval $[1,2].
For the Euler method this excludes step sizes $\Delta x>0.05$, as that reaches negative values in the first step. For the RK4 method one can compute that the first step reaches negative values in the first step for $Δx>0.096927124$, however the following steps still could reach below the axis in the further steps.
Numerical experimentation gives the following plots and critical step sizes.