Final Answer:
The answer to the given equation is: (x^3)/3 + (x^2)y - xy^2 - y^2/2 = c
Step-by-step explanation:
The equation (xy2+y-x)dx+x(xy+1)dy=0 is a first order linear differential equation. It can be solved by the method of integrating factors. To solve this equation, we need to multiply the equation by an integrating factor, which is a function of x and y alone.
The integrating factor is defined as exp[∫P(x, y)dx], where P(x, y) is the coefficient of dx in the given equation. So, we get the integrating factor as exp[∫x(xy+1)dx]. By integrating x(xy+1), we get the integrating factor as exp[(x^3)/3 + (x^2)y - xy^2].
Now, multiplying the given equation by the integrating factor, we get: (xy2+y-x)*[exp[(x^3)/3 + (x^2)y - xy^2]]dx + x(xy+1)*[exp[(x^3)/3 + (x^2)y - xy^2]]dy = 0.
Integrating both sides with respect to y, we get: (x^3)/3 + (x^2)y - xy^2 - y^2/2 = c, where c is an arbitrary constant.
This is the solution to the given equation. The left-hand side of the equation is a function of x and y, and the right-hand side consists of an arbitrary constant. Thus, we can conclude that the equation (xy2+y-x)dx+x(xy+1)dy=0 is solved by integrating both sides with respect to y.