Solving Differential Equations of the Form ( frac{dy}{dx} P(x)y Q(x) ): A Step-by-Step Guide

Introduction

In the field of mathematics, differential equations are central to modeling complex systems in various scientific and engineering disciplines. One common form of these equations is ( frac{dy}{dx} P(x)y Q(x) ). This article will walk through the process of solving such equations using the integrating factor method, providing a clear and detailed explanation.

Understanding the Given Problem

Consider the differential equation:

$$ x^2 1 frac{dy}{dx} - 4xy frac{1}{x^3} 1 $$

To solve this, we first rewrite it in the standard form ( frac{dy}{dx} P(x)y Q(x) ):

$$ frac{dy}{dx} - frac{4xy}{x^2 1} frac{1}{x^3} 1 $$

Identifying ( P(X) ) and ( Q(X) )

From the standard form, we can identify ( P(x) ) and ( Q(x) ) as:

$$ P(x) -frac{4x}{x^2 1} $$

$$ Q(x) frac{1}{x^3} 1 $$

The Integrating Factor (I.F.) Method

The integrating factor method involves the following steps:

Step 1: Calculate the Integrating Factor (I.F.)

The I.F. is given by the exponential of the integral of ( P(x) ):

$$ I.F. e^{int P(x) , dx} $$

Substituting ( P(x) -frac{4x}{x^2 1} ):

$$ I.F. e^{int -frac{4x}{x^2 1} , dx} $$

We can simplify the integral:

$$ int -frac{4x}{x^2 1} , dx -2 int frac{2x}{x^2 1} , dx $$

Let ( u x^2 1 ), then ( du 2x , dx ):

$$ -2 int frac{1}{u} , du -2 ln |u| -2 ln |x^2 1| ln (x^2 1)^{-2} $$

Thus, the integrating factor is:

$$ I.F. e^{ln (x^2 1)^{-2}} (x^2 1)^{-2} frac{1}{(x^2 1)^2} $$

Step 2: Apply the Integrating Factor to the Equation

Multiplying both sides of the differential equation by the I.F., we get:

$$ frac{1}{(x^2 1)^2} cdot frac{dy}{dx} - frac{4xy}{x^2 1} cdot frac{1}{(x^2 1)^2} frac{1}{x^3 1} cdot frac{1}{(x^2 1)^2} frac{1}{(x^2 1)^2} $$

$$ frac{1}{(x^2 1)^2} cdot frac{dy}{dx} - frac{4xy}{(x^2 1)^3} frac{1}{(x^3 1)(x^2 1)^2} frac{1}{(x^2 1)^2} $$

This can be simplified to:

$$ frac{1}{(x^2 1)^2} cdot frac{dy}{dx} frac{d}{dx} left( -frac{2y}{(x^2 1)} right) frac{1}{x^2 1} $$

Step 3: Integrate Both Sides

Using the linearity property of the integral, we integrate:

$$ y (x^2 1)^{-2} int frac{1}{x^2 1} , dx C $$

The integral of ( frac{1}{x^2 1} ) is ( tan^{-1} x ):

$$ y (x^2 1)^{-2} tan^{-1} x C $$

Step 4: Solve for ( y )

Multiplying both sides by ( (x^2 1)^2 ), we get the general solution:

$$ y (x^2 1)^2 left( tan^{-1} x C right) $$

Conclusion

Solving differential equations of the form ( frac{dy}{dx} P(x)y Q(x) ) using the integrating factor method is a systematic and powerful approach. By identifying ( P(x) ) and ( Q(x) ) and calculating the integrating factor, we can transform the equation into a more manageable form, allowing us to find the general solution.