Direct integration initial value problems

#045

Problem

y' = 2x\sin(x^2),\quad y(0)=3

Classification

trigonometric antiderivative
oscillation
initial condition

Method

direct integration
initial value problem
integration substitution

Solution

y = -\cos(x^2) + 4

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-4, 4, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = -np.cos(x**2) + 4
    y_prime = 2*x*np.sin(x**2)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = -cos(x^2) + 4", linewidth=2)
plt.plot(x, y_prime, label="y' = 2xsin(x^2)", linewidth=2, color="red")
plt.title("Entry #045: 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 2xsin(x^2). After using integration, we can pinpoint the solution because the initial condition tells us y(0)=3.