Solving the Differential Equation (yy^21 dx xy^2 - 1 dy 0) Using Inspection and Integration Techniques

Understanding and solving differential equations is a critical skill in applied mathematics and engineering. This article focuses on solving a particular differential equation using both inspection and integration techniques. The equation in question is (yy^{21} dx (xy^2 - 1) dy 0). We will explore step-by-step methods to solve this equation, including the application of the integrating factor method, decomposition into partial fractions, and integration.

Introduction to the Problem

The given differential equation is:

[ yy^{21} dx (xy^2 - 1) dy 0 ]

We can separate the variables to analyze the nature of the equation. However, recognizing that it is not homogeneous and exploring whether it is separable can provide insights into solving it.

Separation of Variables

First, we separate the variables (y) and (x) by moving all terms involving (y) to one side and all terms involving (x) to the other. The given equation can be reorganized as:

[ dy -frac{yy^{21}}{xy^2 - 1} dx ]

Manipulating the Equation

To simplify the right-hand side, we start by rewriting the term (frac{xy^2 - 1}{yy^{21}}):

[ frac{y^2 - 1}{yy^{21}} frac{2y^2 - y^2 - 1}{yy^{21}} frac{2y^2}{yy^{21}} - frac{y^21}{yy^{21}} frac{2y}{y^{21}} - frac{1}{y} ]

Thus, the integral equation becomes:

[ int left( frac{2y}{y^{21}} - frac{1}{y} right) dy int -frac{1}{x} dx ]

Next, we integrate both sides of the equation:

[ int left( frac{2y}{y^{21}} - frac{1}{y} right) dy int -frac{1}{x} dx ]

The left-hand side integrates to:

[ ln(y^{21}) - ln(y) ln left( frac{y^{21}}{y} right) ln left( y^{20} right) ]

The right-hand side integrates to:

[ -ln(x) C ]

So, we have:

[ ln left( frac{y^{20}}{x} right) C ]

Exponentiating both sides, we get:

[ frac{y^{20}}{x} k ]

Finally, solving for (y):

[ y^{20} kx ]

Thus, the general solution is:

[ y left( kx right)^{frac{1}{20}} ]

Verification and Alternative Solution

An alternative approach involves using the method of partial fractions. We can decompose the left-hand side as follows:

[ frac{2y}{y^{21}} - frac{1}{y} frac{2}{y^{20}} - frac{1}{y} ]

Integrating both sides:

[ int left( frac{2}{y^{20}} - frac{1}{y} right) dy int -frac{1}{x} dx ]

Which simplifies to:

[ -frac{2}{19} cdot frac{1}{y^{19}} - ln(y) -ln(x) C ]

Reorganizing, we get:

[ -frac{2}{19} cdot frac{1}{y^{19}} - ln(y) ln(x) C ]

Exponentiating both sides and solving for (y), we obtain a similar form:

[ y left( kx right)^{frac{1}{20}} ]

Conclusion

The differential equation (yy^{21} dx (xy^2 - 1) dy 0) can be solved using separation of variables and integration techniques. The general solution is ( y left( kx right)^{frac{1}{20}} ), where (k) is an arbitrary constant. This solution demonstrates the application of both direct integration and the method of partial fractions in solving complex differential equations.

Keywords: Differential Equations, Integrating Factor, Separable Equations, Partial Fractions