Back to archive

Direct integration equations

#002

Problem

y′=3x2y' = 3x^2

Classification

  • power rule
  • solution family

Method

  • direct integration

Solution

y=x3+Cy = x^3 + C

Simulation & Code

Computational workSimulation
Graph visualization for entry #002
Download .py file
Python codeOpen
import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-3, 3, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = x**3
    y_prime = 3*x**2

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = x^3", linewidth=2)
plt.plot(x, y_prime, label="y' = 3x^2", linewidth=2, color="red")
plt.title("Entry #002: solution and derivative")
plt.xlabel("x")
plt.ylabel("value")
plt.grid(True, alpha=0.35)
plt.legend()
plt.tight_layout()
plt.show()

Handwritten derivation

Download full PDF
PDF preview unavailable. Use the download link above to view the handwritten work.

Takeaway & Interpretation

This equation says the solution's slope is controlled entirely by 3x^2. After integrating, the solution family becomes y = x^3 + C, meaning every solution has the same overall shape but may be shifted vertically by the constant C.