1. Problem 1: Find the integrating factor for the differential equation $$dx + \left(\frac{x}{y} - \sin y\right) dy = 0.$$ - Generally, the integrating factor depends on whether the equation is exact or can be made exact by multiplying by a function of $x$ or $y$. - The options given are: A. $x$ B. $y$ C. $x^{2}y^{3}$ D. $\frac{x^{2}}{y^{3}}$ - By analyzing the form, the integrating factor is $y$ (Option B) to make this equation exact. 2. Problem 2: Find the integrating factor for the differential equation $$y(x + y)dx + (x + 2y - 1)dy = 0.$$ - The options are: A. $\ln x$ B. $e^{x}$ C. $x$ D. $1$ - Testing reveals that the integrating factor is $x$ (Option C). 3. Problem 3: Find the solution for the differential equation $$y(x + y)dx + (x + 2y - 1)dy = 0.$$ - Options: A. $y(x + y - 1) = ce^{-x}$ B. $c = (x - 1)y + y^{2}e^{x}$ C. $y(x + y - 1) = ce^{x}$ D. $c = x^{2}(x - 1)y + y^{2}e^{x}$ - The correct solution is $y(x + y - 1) = ce^{-x}$ (Option A). Final answers: - Problem 1: B ($y$) - Problem 2: C ($x$) - Problem 3: A ($y(x + y - 1) = ce^{-x}$)