Exponential Decay

Exponential Decay

The simplest real-world ODE:

$\frac{dy}{dt} = -k \cdot y, \quad y(0) = y_0$

The rate of change is proportional to the current value. The exact solution is:

$y(t) = y_0 \cdot e^{-kt}$

Applications

• Radioactive decay: atoms decay at a rate proportional to their count
• Drug clearance: concentration in blood decreases proportionally
• Capacitor discharge: voltage drops exponentially
• Population decline: a species dying off at a fixed rate

Half-Life

The half-life is the time for $y$ to reach half its initial value:

$\frac{y_0}{2} = y_0 \cdot e^{-k \cdot t_{\text{half}}} \implies t_{\text{half}} = \frac{\ln 2}{k} \approx \frac{0.693}{k}$

Numerical Solution

Using Euler's method with $n$ steps:

def exponential_decay(k, y0, t_end, n):
    h = t_end / n
    y = float(y0)
    for _ in range(n):
        y = y + h * (-k * y)
    return y

Your Task

Implement exponential_decay(k, y0, t_end, n) that numerically solves $\frac{dy}{dt} = -k \cdot y$ and returns the final value $y(t_{\text{end}})$.