a very interesting differential equation

This paragraph introduces the differential equation \( y \cdot y' = (y')^2 \), highlighting its nonlinearity due to the quadratic nature of the term involving \( y' \). The discussion covers the equation's classification as autonomous and nonlinear, and an initial approach to solving it by substitution. The author then introduces a new variable \( z = y' \), transforming the equation into a second-order nonlinear equation. A key insight involves recognizing a pattern resembling the chain rule, allowing further simplification.

The author proceeds with the solution by applying the method of separation of variables. The transformed equation \( 2 \cdot \frac{dz}{dx} = z^2 \) is separated into terms involving \( z \) and \( x \). After integrating both sides, the equation is simplified, yielding an expression for \( z \), which corresponds to \( y' \). The paragraph concludes by finding the antiderivative of this expression to derive a function for \( y \), completing the solution for the case when the constant \( a \) is zero.

In this paragraph, the author tackles the more complex scenario where the constant \( a \) is positive. Using separation of variables again, the differential equation is transformed and integrated. The antiderivative for the left-hand side involves the arctangent function, and after some algebraic manipulation, the equation for \( z \) (which equals \( y' \)) is obtained. Finally, the author derives the solution for \( y \) by integrating the obtained expression for \( y' \), resulting in a solution involving the natural logarithm of the secant function.

This paragraph focuses on the scenario where \( a \) is positive, but the differential equation involves a minus sign. The author begins by separating the variables and then addresses the integral on the left side through partial fraction decomposition. After deriving the antiderivative, the equation is further simplified by exponentiating both sides. The solution is expressed in terms of \( Z \) and eventually transformed back to \( y' \). The paragraph concludes by presenting the integrated solution for \( y \), which includes a logarithmic expression.

In this concluding paragraph, the author combines the solutions from the previous cases. By further simplifying the expression for \( y \), involving natural logarithms and exponentials, the final form of the solution is derived. The author manipulates the equation to achieve a more elegant expression, suitable for the original differential equation. This final step completes the discussion and provides a neat, closed-form solution to the complex nonlinear differential equation introduced at the beginning.