Solving the Differential Equation dy/dx (y-1)/(x(y-1))

In this article, we will discuss how to solve the differential equation (frac{dy}{dx} frac{y-1}{x(y-1)}). This type of equation can be quite complex, but we will break it down into manageable steps, including separation of variables and the use of integrating factors.

Introduction to the Equation

Consider the differential equation (frac{dy}{dx} frac{y-1}{x(y-1)}). The goal is to find a function (y(x)) that satisfies this equation for all (x). This is a nonlinear first-order differential equation, and we will explore methods to solve it step-by-step.

Method of Separation of Variables

To start, let's try the method of separation of variables. We rewrite the equation in a form that separates the dependent and independent variables:

(frac{dy}{y-1} frac{dx}{x})

Integrating both sides, we get:

(int frac{dy}{y-1} int frac{dx}{x})

The integrals on both sides are:

(ln|y-1| ln|x| C)

Exponentiating both sides:

(|y-1| C|x|)

This can be written as:

(y-1 pm Cx)

Hence, the general solution is:

(y Cx 1)

Substitution Method

Alternatively, we can use a substitution to simplify the equation. Let's substitute:

(v y - 1)

Then, (y v 1) and (dy dv). Substituting these into the original equation:

(frac{dv}{dx} frac{v}{xv} frac{1}{x})

Separating variables gives:

(v dv frac{dx}{x})

Integrating both sides:

(frac{v^2}{2} ln|x| C)

Substituting back for (v):

(left(y-1right)^2 2ln|x| C)

Thus, the general solution is:

(y 1 sqrt{2ln|x| C})

Linear Differential Equation and Integrating Factor

Another approach is to consider the equation as a linear differential equation. We can write:

(frac{dy}{dx} - frac{y-1}{x} 0)

Here, the integrating factor (mu(x)) is given by:

(mu(x) expleft(int -frac{1}{x} dxright) exp(-ln|x|) frac{1}{x})

Multiplying the entire equation by the integrating factor:

(frac{1}{x} cdot frac{dy}{dx} - frac{y-1}{x^2} 0)

The left side can be written as:

(frac{d}{dx} left(frac{y-1}{x}right) 0)

Integrating both sides:

(frac{y-1}{x} C)

Hence:

(y Cx 1)

Reformulation and Reciprocation

Rewriting the equation as (frac{dy}{dx} frac{1}{x}) to solve for (xy), we reciprocate both sides:

(frac{dx}{dy} frac{x}{y-1})

Introducing a new variable (v y-1) and using the integrating factor method again:

(frac{dv}{dx} - frac{v}{x} 1)

The integrating factor is:

(mu(x) expleft(int -frac{1}{x} dxright) frac{1}{x})

Multiplying the equation by the integrating factor:

(frac{1}{x} cdot frac{dv}{dx} - frac{v}{x^2} frac{1}{x})

The left side can be written as:

(frac{d}{dx} left(frac{v}{x}right) frac{1}{x})

Integrating both sides:

(frac{v}{x} ln|x| C)

Thus:

(v Cx xln|x|)

Substituting back:

(y - 1 Cx xln|x|)

Therefore, the general solution is:

(y 1 Cx xln|x|)

Conclusion

In this article, we have explored multiple methods to solve the differential equation (frac{dy}{dx} frac{y-1}{x(y-1)}). The solutions include using separation of variables, substitution, and integrating factors. Each method provides a different perspective on solving the equation, and understanding multiple approaches can be helpful in tackling similar problems in the future.

Keywords

Keywords: differential equation, separable equation, integrating factor