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Direct integration equations

#003

Problem

y′=5x4−2x+7y' = 5x^4 - 2x + 7Open Model Lab

Classification

  • power rule
  • solution family

Method

  • direct integration

Solution

y=x5−x2+7x+Cy = x^5 - x^2 + 7x + C

Simulation & Code

Computational workSimulation
Graph visualization for entry #003
Download .py file
Python codeOpen
import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-2, 2, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = x**5 - x**2 + 7*x
    y_prime = 5*x**4 - 2*x + 7

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = x^5 - x^2 + 7x", linewidth=2)
plt.plot(x, y_prime, label="y' = 5x^4 - 2x + 7", linewidth=2, color="red")
plt.title("Entry #003: solution and derivative")
plt.xlabel("x")
plt.ylabel("value")
plt.grid(True, alpha=0.35)
plt.legend()
plt.tight_layout()
plt.show()

Handwritten derivation

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Takeaway & Interpretation

This equation says the solution's slope is controlled entirely by 5x^4 - 2x + 7. After integrating, the solution family becomes y = x^5 - x^2 + 7x + C, meaning every solution has the same overall shape but may be shifted vertically by the constant C.