Direct integration initial value problems

#047

Problem

y' = \frac{1}{1 + 4x^2},\quad y(0)=-1

Classification

inverse trig pattern
initial condition

Method

direct integration
initial value problem
integration substitution

Solution

y = \frac{1}{2}\arctan(2x) - 1

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

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

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = 0.5*np.arctan(2*x) - 1
    y_prime = 1/(1 + 4*x**2)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = (1/2)arctan(2x) - 1", linewidth=2)
plt.plot(x, y_prime, label="y' = 1/(1 + 4x^2)", linewidth=2, color="red")
plt.title("Entry #047: 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/(1 + 4x^2). After using integration, we can pinpoint the solution because the initial condition tells us y(0)=-1.