Final answer:
To find the integrating factor for the differential equation, one must manipulate it into an exact equation by determining a function μ(x) or μ(y). Calculate partial derivatives of M and N to find the integrating factor that makes the equation exact, and then solve the resulting equation.
Step-by-step explanation:
To find an integrating factor for the given differential equation, we can try to manipulate it into an exact equation. The differential equation presented is (xy + x + 2y + 1) dx + (x + 1) dy = 0. To find an integrating factor, μ, which depends either solely on x or y (denoted as μ(x) or μ(y)), we will determine if our equation can be made exact by finding a function μ that when multiplied to both sides makes the mixed partial derivatives of the resulting P(x, y) and Q(x, y) equal.
In this case, we will look for an integrating factor of the form μ(x) since the terms involving y seem to be simpler. For an equation of the form M(x, y)dx + N(x, y)dy = 0, where our M is (xy + x + 2y + 1) and N is (x + 1), an integrating factor μ(x) exists if μ'(x)/μ(x) = (N_y - M_x) / M, where M_x and N_y are the partial derivatives with respect to x and y, respectively.
By calculating the partial derivatives and substituting into this formula, we can solve the resulting differential equation to find the integrating factor, which can then be applied to the original equation to solve it.