Final Answer:
The differential equation dy/dx + x/(x² + 1)y = xy² is a first-order linear ordinary differential equation.
Step-by-step explanation:
The provided differential equation is expressed as dy/dx + P(x)y = Q(x)y², where P(x) = x/(x² + 1) and Q(x) = x. This configuration characterizes a first-order linear ordinary differential equation due to its linearity in y and its derivative, while also being of the first order.
Typically, the solution to such a differential equation involves employing an integrating factor, denoted by e^(∫P(x)dx). For the given equation, the integrating factor becomes e^(∫x/(x² + 1)dx). Upon integration, this results in (1/2)ln|x² + 1|, and subsequently, the integrating factor is √(x² + 1).
Multiplying both sides of the differential equation by the integrating factor leads to √(x² + 1)dy/dx + x/√(x² + 1)y = xy², which simplifies to the exact differential form d/dx(y√(x² + 1)) = xy².
In conclusion, the differential equation provided is indeed a first-order linear ordinary differential equation, and the solution involves utilizing an integrating factor to transform it into an exact differential form