Direct integration initial value problems

#031

Problem

y' = \frac{1}{\sqrt{x}},\quad y(1)=4

Classification

power rule
domain restriction
initial condition

Method

direct integration
initial value problem

Solution

y = 2\sqrt{x} + 2

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) + 2
    y_prime = 1/np.sqrt(x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = 2sqrt(x) + 2", linewidth=2)
plt.plot(x, y_prime, label="y' = 1/sqrt(x)", linewidth=2, color="red")
plt.title("Entry #031: 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

The differential equation tells us that the slope is 1/sqrt(x). After using integration, we can pinpoint the solution because the initial condition tells us y(1)=4. This differential equation only takes positive values, this means the slope of the solution is always positive.