Solving Differential Equations of the Form ( frac{dy}{dx} frac{yx - y}{x^2 y} )

In this article, we will explore how to solve a specific type of first-order differential equation. The differential equation we consider is:

Problem: Solve the differential equation (frac{dy}{dx} frac{yx - y}{x^2 y}).

Method 1: Substitution with y vx

Let's start by making the substitution (y vx). Differentiating both sides with respect to (x), we get:

[frac{dy}{dx} v xfrac{dv}{dx}]

Substituting (y vx) and (frac{dy}{dx} v xfrac{dv}{dx}) into the original equation, we obtain:

[v xfrac{dv}{dx} frac{vx - v}{x^2 vx}]

Dividing through by (v) gives:

[1 xfrac{dv}{dx} frac{1 - frac{1}{v}}{1 frac{1}{v}} frac{1 - v}{1 v}]

Thus, we have:

[xfrac{dv}{dx} frac{1 - v}{1 v} - 1]

Which simplifies to:

[frac{1}{1 - 2v - v^2}frac{dv}{dx} -frac{2}{x}]

Integrating both sides, we get:

[-frac{1}{2}intfrac{1 - 2v - v^2}{1 - 2v - v^2}dv -2intfrac{1}{x}dx]

This leads to:

[-frac{1}{2} ln|1 - 2v - v^2| -2 ln|x| C]

and thus:

[ln|sqrt{1 - 2v - v^2}| 2 ln|x| - C]

which simplifies to:

[frac{1}{sqrt{1 - 2v - v^2}} Cx^2]

Substituting back (v frac{y}{x}), we get:

[frac{1}{sqrt{1 - 2frac{y}{x} - left(frac{y}{x}right)^2}} Cx^2]

The final solution is then:

[y x sqrt{frac{C}{x^2} - 1}]

or equivalently:

[y^2 Cx^2 - x^2]

Conclusion

Through the application of substitution and integration techniques, we have successfully found the general solution to the given differential equation. These methods highlight the power of different substitutions in solving complex differential equations, and the importance of maintaining conceptual clarity in each step.

For more detailed insights into differential equations and advanced calculus, further mathematical exploration and practice is highly recommended.