Direct integration equations

#006

Problem

y' = \frac{1}{\sqrt{x}}

Classification

power rule
domain restriction
solution family

Method

direct integration

Solution

y = 2\sqrt{x} + C

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0.03, 9, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = 2*np.sqrt(x)
    y_prime = 1/np.sqrt(x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = 2sqrt(x)", linewidth=2)
plt.plot(x, y_prime, label="y' = 1/sqrt(x)", linewidth=2, color="red")
plt.title("Entry #006: 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 1/sqrt(x). After integrating, the solution family becomes y = 2sqrt(x) + 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. Also, y' blows up at x=0, meaning the solution develops a vertical-tangent-type behavior near that point.