Direct integration equations

#005

Problem

y' = 4\sqrt{x}

Classification

power rule
solution family

Method

direct integration

Solution

y = \frac{8}{3}x^{3/2} + C

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 6, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = (8/3)*np.power(x, 1.5)
    y_prime = 4*np.sqrt(x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = (8/3)x^(3/2)", linewidth=2)
plt.plot(x, y_prime, label="y' = 4sqrt(x)", linewidth=2, color="red")
plt.title("Entry #005: 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 4sqrt(x). After integrating, the solution family becomes y = (8/3)x^(3/2) + C, meaning every solution has the same overall shape but may be shifted vertically by the constant C. This function only takes on positive values, so the solution is increasing everywhere on its domain.