Direct integration initial value problems

#049

Problem

y' = e^x + \cos x + 2x,\quad y(0)=0

Classification

exponential antiderivative
trigonometric antiderivative
power rule
initial condition

Method

direct integration
initial value problem

Solution

y = e^x + \sin x + x^2 - 1

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) + np.sin(x) + x**2 - 1
    y_prime = np.exp(x) + np.cos(x) + 2*x

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = e^x + sin(x) + x^2 - 1", linewidth=2)
plt.plot(x, y_prime, label="y' = e^x + cos(x) + 2x", linewidth=2, color="red")
plt.title("Entry #049: 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 e^x + cos(x) + 2x. After using integration, we can pinpoint the solution because the initial condition tells us y(0)=0.