A differential equation specifies local slope, and a numerical method follows that slope in finite steps.
WHAT HAPPENS IF…For λ=1, y₀=1, h=0.5, what is the four-step Euler value at t=2?
Interactive exploration
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Connect the picture to the ideay′=λy,y(t)=y0eλt,yk+1=(1+λh)yk
Euler’s method uses the current derivative to advance h units in time. The exact exponential and the discrete multiplier 1+λh agree to first order in h. For negative λ, a large h can produce oscillation or growth even though the exact solution decays.
A detail that matters
Numerical instability belongs to the method, not necessarily to the differential equation.