Direct integration initial value problems

#037

Problem

y' = 4\cos x,\quad y(0)=-1

Classification

trigonometric antiderivative
oscillation
initial condition

Method

direct integration
initial value problem

Solution

y = 4\sin x - 1

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 = 4*np.sin(x) - 1
    y_prime = 4*np.cos(x)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = 4sin(x) - 1", linewidth=2)
plt.plot(x, y_prime, label="y' = 4cos(x)", linewidth=2, color="red")
plt.title("Entry #037: solution and derivative")
plt.xlabel("x")
plt.ylabel("value")
plt.grid(True, alpha=0.35)
plt.legend()
plt.tight_layout()
plt.show()

Handwritten derivation

Download full PDF

Takeaway & Interpretation

The differential equation tells us that the slope is 4cos(x). After using integration, we can pinpoint the solution because the initial condition tells us y(0)=-1.