Direct integration initial value problems

#036

Problem

y' = \sin x,\quad y(0)=2

Classification

trigonometric antiderivative
oscillation
initial condition

Method

direct integration
initial value problem

Solution

y = -\cos x + 3

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-7, 7, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = -np.cos(x) + 3
    y_prime = np.sin(x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = -cos(x) + 3", linewidth=2)
plt.plot(x, y_prime, label="y' = sin(x)", linewidth=2, color="red")
plt.title("Entry #036: 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 sin(x). After using integration, we can pinpoint the solution because the initial condition tells us y(0)=2.