Solving the Differential Equation y xy^3 - xy: A Comprehensive Guide

In this article, we will delve into the process of solving a specific type of differential equation, namely y xy^3 - xy. This equation is nonlinear and falls under the category of Bernoulli equations. We will explore the step-by-step method to solve it, including the use of partial fractions and integrating factors.

Introduction to the Differential Equation

The given differential equation is:

dy/dx xy^3 - xy

Transforming the Equation

First, we can rewrite the equation as:

dy/dx - xy xy^3

This form shows that it's a nonlinear first-order differential equation, specifically a Bernoulli equation with k 3.

Solving the Differential Equation

Substitution Method

To solve this equation, we can use the substitution method. Let's set:

z y-2

Then:

dz/dx -2y-3dy/dx

Multiplying the original differential equation by -2y-3, we get:

-2y-3dy/dx - 2xy-2 -2x

Substituting z and dz/dx into the equation:

dz/dx - 2xz -2x

Using an Integrating Factor

We can now use the integrating factor method. The integrating factor is:

μ e∫2xdx ex2

Multiplying the entire equation by the integrating factor:

ex2dz/dx - 2xex2z -2xex2

This can be rewritten as:

d/dx[ex2z] -2xex2

Integrating Both Sides

Integrating both sides:

ex2z -∫2xex2dx

We can use u-substitution with u x2, so du 2xdx. Then the integral becomes:

ex2z -∫eudu -eu C -ex2 C

Substituting back z y-2:

ex2y-2 Ce-x2

Therefore:

y2 1/Cex2

Alternative Method: Partial Fractions

Another approach to solving this equation is to use partial fractions. Starting from:

dy/dx xy^3 - y

We can rewrite the equation as:

dy/(y^3 - y) xdx

The integrand on the left side can be decomposed using partial fractions:

1/(y(y - 1)(y 1)) A/y B/(y - 1) C/(y 1)

Using the cover-up method, we can find:

A -1, B 1/2, C 1/2

Thus the integral becomes:

∫(1/(y(y - 1)(y 1)))dy -∫(1/y)dy (1/2)∫(1/(y - 1))dy (1/2)∫(1/(y 1))dy

Evaluating these integrals:

-ln|y| (1/2)ln|y - 1| (1/2)ln|y 1| (1/2)x^2 C

Combining the logarithms and applying the exponential function:

ln|((y - 1)(y 1))/(y)| (1/2)x^2 C

This simplifies to:

|((y^2 - 1)/y)| Ce^(x^2/2)

Conclusion

In summary, we have explored two methods for solving the differential equation y xy^3 - xy. Both methods involve transforming the original equation into a more manageable form and then applying integration techniques such as partial fractions and integrating factors. Understanding these methods is crucial for solving nonlinear differential equations and enhancing problem-solving skills in advanced calculus.