Direct integration initial value problems

#044

Problem

y' = 5\cos(5x),\quad y(0)=2

Classification

trigonometric antiderivative
oscillation
initial condition

Method

direct integration
initial value problem

Solution

y = \sin(5x) + 2

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-2.5, 2.5, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = np.sin(5*x) + 2
    y_prime = 5*np.cos(5*x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = sin(5x) + 2", linewidth=2)
plt.plot(x, y_prime, label="y' = 5cos(5x)", linewidth=2, color="red")
plt.title("Entry #044: 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 5cos(5x). After using integration, we can pinpoint the solution because the initial condition tells us y(0)=2.