Solving the Differential Equation y' xy - 1

In this article, we will explore the solution to the differential equation (frac{dy}{dx} xy - 1). This type of equation is a first-order nonlinear ordinary differential equation (ODE) and provides an interesting framework to discuss integration and singular solutions.

Step-by-Step Integration

The given differential equation is:

(frac{dy}{dx} xy - 1)

First, we can rewrite the equation as:

(frac{dy}{y-1} xdx)

To integrate both sides, we start with the left side. Let (u y - 1). Then, (du dy), and we can rewrite the integral as:

Converting back to (y), we get:

(ln|y-1| C_1 ln|y-1| C)

For the right side, the integral is simple:

Combining both sides, we have:

(ln|y-1| C_1 frac{1}{2}x^2 C_2)

Cancelling the constants, we can write:

(ln|y-1| frac{1}{2}x^2 C)

Exponentiating both sides to remove the natural logarithm:

(|y-1| e^{frac{1}{2}x^2 C})

Simplifying, we obtain:

(y-1 C'e^{frac{x^2}{2}}) where (C' e^C)

Adding 1 to both sides, we can express the solution as:

(y 1 C'e^{frac{x^2}{2}})

Determining the Particular Integral

A particular integral is a specific solution that can be found by assigning a constant value to (C'). In this case, the solution provided is:

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

Substituting this back into the differential equation:

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

And:

(xy - 1 xleft(frac{1}{1-x}right) - 1 1)

This confirms that (y frac{1}{1 - x}) is indeed a particular solution.

Conclusion

The differential equation (frac{dy}{dx} xy - 1) has been solved, and a particular integral has been found. The general solution is given by:

(y 1 C'e^{frac{x^2}{2}})

Additionally, a specific solution for (C' -frac{1}{1-x}) is:

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

This problem demonstrates the integration techniques for nonlinear ODEs and the process of finding particular and general solutions.