Solving the Differential Equation y' y sin(x)cos(x) e^{sin^2(x)}

Introduction

Differential equations are a fundamental part of calculus, often encountered in various scientific and engineering applications. This article focuses on solving a specific differential equation:

Solution of the Differential Equation

Let's begin with the differential equation:

y' y sin(x)cos(x) e^{sin^2(x)}

To solve this, we can use an integrating factor, a technique that turns the left-hand side of the equation into a derivative of a product.

Step 1: Find the Integrating Factor

The first step is to find a special function mu; called the “integrating factor” with the following property:

-sin(x)cos(x)mu; frac;dmu;{dx}

To find such a function, we integrate:

int -sin(x)cos(x)dx int frac;dmu{mu}

This leads to:

- frac{1}{2}sin^2(x) ln(mu) Const

Thus, we have:

mu; Ae^{-sin^2(x)/2}

Step 2: Multiply the Equation by the Integrating Factor

We multiply the entire differential equation by this integrating factor:

y'mu; y sin(x)cos(x)mu; e^{sin^2(x)}mu;

Using the property of the integrating factor:

-sin(x)cos(x)mu; frac{dmu;}{dx}

The left-hand side can be rewritten as:

ye^{sin^2(x)/2}

The equation now becomes:

ye^{sin^2(x)/2} frac{d}{dx}ye^{sin^2(x)/2}

Integrating both sides:

ye^{sin^2(x)/2} e^{sin^2(x)/2}int_{x_0}^{x}e^{sin^2(t)/2}dt

Simplifying, we get:

y e^{-sin^2(x)/2}int_{x_0}^{x}e^{sin^2(t)/2}dt Ce^{-sin^2(x)/2}

Where C is a constant of integration, and x_0 is the initial condition.

General Solution

The general solution is then:

y(x) e^{sin^2(x)/2}cdot[int e^{sin^2(t)/2}dt C]

For practical purposes, the integral int e^{sin^2(t)/2}dt is often expressed as a series expansion, as given in the initial solution.

Conclusion

By carefully selecting an integrating factor and integrating the modified equation, we have arrived at a solution that fits the original differential equation.

For a more detailed understanding, consider reviewing Pauls Online Notes on Linear Differential Equations.

Key Points:

Integration Factor: A special function used to simplify the process of solving linear differential equations. Linear Differential Equation: A differential equation where the dependent variable and its derivatives appear only to the first power. Initial Condition: The value of the function at a specified point, often denoted by x_0.

Understanding these concepts is essential for solving differential equations and their applications in various scientific fields.