Solution of the differential equation y d y d x +x=0 represents a family of

Step-by-step solution

The given differential equation is y dy dx + x = 0 . Rearranging the terms, we have y dy dx = -x . Integrating both sides with respect to x , we get y , dy = - x , dx . This yields y^2 2 = - x^2 2 + C , where C is the constant of integration. Multiplying by 2, we obtain y^2 = -x^2 + 2C , which simplifies to x^2 + y^2 = 2C . Let 2C = R^2 , where R^2 > 0 . Then x^2 + y^2 = R^2 . This is the standard equation of a family of circles centered at the origin (0,0) with radius R . Answer: Circles