Direct integration initial value problems

#032

Problem

y' = 6x^2 - 4x^{-1},\quad y(1)=0

Classification

power rule
logarithmic antiderivative
initial condition

Method

direct integration
initial value problem

Solution

y = 2x^3 - 4\ln|x| - 2

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-3, 3, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = 2*x**3 - 4*np.log(np.abs(x)) - 2
    y_prime = 6*x**2 - 4/x


def mask(values, limit=100):
    return np.where(np.isfinite(values) & (np.abs(values) <= limit), values, np.nan)

y = mask(y)
y_prime = mask(y_prime)
plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = 2x^3 - 4ln|x| - 2", linewidth=2)
plt.plot(x, y_prime, label="y' = 6x^2 - 4x^(-1)", linewidth=2, color="red")
plt.title("Entry #032: solution and derivative")
plt.xlabel("x")
plt.ylabel("value")
plt.grid(True, alpha=0.35)
plt.legend()
plt.tight_layout()
plt.show()

Handwritten derivation

Download full PDF

Takeaway & Interpretation

The differential equation tells us that the slope is 6x^2 - 4x^-1. After using integration, we can pinpoint the solution because the initial condition tells us y(1)=0. This differential equation has a singularity at x=0.