Implicit Euler for y' + a y = 0

1. Solution plot (T = 1)

2. Convergence (T = 1)

3. Stability Demonstration (T = 10)

This plot compares the behavior of Implicit and Explicit Euler methods using a large time step (h=1.1) that violates the stability condition for Explicit Euler (h < 1.0). Notice how the Implicit Euler solution remains stable (decays), while the Explicit Euler solution blows up.