Direct integration initial value problems

#040

Problem

y' = 4x^3 - 6x + 2,\quad y(1)=0

Classification

power rule
initial condition

Method

direct integration
initial value problem

Solution

y = x^4 - 3x^2 + 2x

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 = x**4 - 3*x**2 + 2*x
    y_prime = 4*x**3 - 6*x + 2

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = x^4 - 3x^2 + 2x", linewidth=2)
plt.plot(x, y_prime, label="y' = 4x^3 - 6x + 2", linewidth=2, color="red")
plt.title("Entry #040: 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 4x^3 - 6x + 2. After using integration, we can pinpoint the solution because the initial condition tells us y(1)=0.