Solving a Non-Exact Differential Equation: A Step-by-Step Guide

In this article, we walk through the process of solving a non-exact differential equation. We'll go through each step of the solution process, from checking for exactness to finding an integrating factor, and finally, integrating both sides to obtain the general solution.

Introduction to the Problem

We begin with the differential equation:

  x^4y^{21}dx   yx^2 - 3x^2 dy  0

Our goal is to find the general solution to this equation. Let's rewrite it in the standard form:

  M(x, y)dx   N(x, y)dy  0

where

  M(x, y)  x^4y^{21}
  N(x, y)  yx^2 - 3x^2

Checking for Exactness

First, we check if the equation is exact.

Determine the partial derivatives of M and N:

  #9166;M/#9166;y  x^4(21y^{20})  21x^4y^{20}
  #9166;N/#9166;x  y(2x - 6x)  y(2x - 6x)  -4yx

Since

  #9166;M/#9166;y  21x^4y^{20} 
eq -4yx  #9166;N/#9166;x,

the equation is not exact. This indicates that we need to find an integrating factor to make the equation exact.

Finding an Integrating Factor

We look for an integrating factor that will make the equation exact. A common approach is to check if the equation can be made exact by multiplying by a function of x or y.

As an alternative, we can rearrange the equation:

  dy/dx  -M/N  -x^4y^{21}/(yx^2 - 3x^2)

This form allows us to separate the variables:

  y dy  -x^4y^{21}/(x^2 - 3x^2) dx

Further simplification of the right-hand side is required:

  x^2 - 3x^2  x^2(1 - 3)  -2x^2

So the equation becomes:

  y dy  x^4y^{21}/(2x^2) dx

Or:

  y dy  x^2y^{21}/2 dx

This can be separated into:

  y dy  (x^2y^{21}/2) dx

Integrating both sides yields:

Separation of Variables and Integration

Integrating the left side:

  int y dy  (y^2)/2   C_1

Integrating the right side involves simplifying:

  int (x^2y^{21}/2) dx

Using the factored form of the denominator, we can express the fraction as partial fractions:

  x^2 - 3x^2  x^2(1 - 3)  -2x^2

Thus:

  -frac{x^4}{x^2 - 3x^2}  A/x   B/x^2

After finding the constants A and B, integrate:

  int (A/x   B/x^2) dx  A ln|x| - B/x   C_2

Combining the results:

  (y^2)/2  A ln|x| - B/x   C

where C C_2 - C_1.

Final Solution

The general solution to the differential equation is:

  (y^2)/2  A ln|x| - B/x   C

This represents the implicit solution to the original differential equation.