Autonomous equations

#052

Problem

y' = -2y

Classification

decay
equilibrium
autonomous

Method

separable

Solution

y = Ce^{-2x}

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-3, 3, 600)
constants = [-4, -2, -1, 1, 2, 4]

fig, ax = plt.subplots(figsize=(7, 4.5))

for constant in constants:
    y = constant * np.exp(-2 * x)
    y = np.where(np.abs(y) <= 8, y, np.nan)
    ax.plot(x, y, linewidth=2, label=f"C = {constant}")

ax.axhline(0, color="black", linewidth=1, alpha=0.7, label="stable equilibrium y = 0")
ax.axvline(0, color="#111827", linewidth=0.8, alpha=0.35)
ax.set_title("Entry #052: y' = -2y")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_ylim(-8, 8)
ax.grid(True, alpha=0.3)
ax.legend(fontsize=8, ncols=2)

fig.tight_layout()
plt.show()

Handwritten derivation

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Takeaway & Interpretation

This is exponential decay with rate 2. Positive solutions decrease toward 0, while negative solutions increase toward 0. The equilibrium y=0 is stable because nearby solutions approach it as x increases.