Direct integration equations

#008

Problem

y' = e^x

Classification

exponential antiderivative
solution family

Method

direct integration

Solution

y = 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 = np.exp(x)
    y_prime = np.exp(x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = e^x", linewidth=2)
plt.plot(x, y_prime, label="y' = e^x", linewidth=2, color="red")
plt.title("Entry #008: 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 e^x. After integrating, the solution family becomes y = e^x + C, meaning every solution has the same overall shape but may be shifted vertically by the constant C. This function is equal to its own derivative, so the nonconstant part of the solution has the same form as the original slope function.