Direct integration equations

#015

Problem

y' = x^3 + e^x

Classification

power rule
exponential antiderivative
solution family

Method

direct integration

Solution

y = \frac{x^4}{4} + e^x + C

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**4/4 + np.exp(x)
    y_prime = x**3 + np.exp(x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = x^4/4 + e^x", linewidth=2)
plt.plot(x, y_prime, label="y' = x^3 + e^x", linewidth=2, color="red")
plt.title("Entry #015: 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 x^3 + e^x. After integrating, the solution family becomes y = x^4/4 + e^x + C, meaning every solution has the same overall shape but may be shifted vertically by the constant C.