(Solved): (a) (6 points) Given the following differential equation, \[ M(x, y)+N(x, y) \frac{\mathrm{d} y}{\ ...

(a) (6 points) Given the following differential equation, \[ M(x, y)+N(x, y) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, \] assume that \( M, N, M_{y} \) and \( N_{x} \) are continuous on \( R:=(\alpha, \beta) \times(\gamma, \delta) \) and that \( M_{y}(x, y)= \) \( N_{x}(x, y) \) for each \( (x, y) \in R \). Fix any \( \left(x_{0}, y_{0}\right) \in R \) and show that \( \psi(x, y) \) is a valid potential, where \[ \psi(x, y):=\int_{x_{0}}^{x} M\left(s, y_{0}\right) \mathrm{d} s+\int_{y_{0}}^{y} N(x, t) \mathrm{d} t . \] (b) (2 points) Show that any separable equation \[ M(x)+N(y) y^{\prime}=0 \] is also exact. (c) (5 points) Show that if \( \left(N_{x}-M_{y}\right) / M=Q \) where \( Q \) is a function of \( y \) only, then the differential equation \[ M(x, y)+N(x, y) \cdot y^{\prime}=0 \] has an integrating factor of the form \[ \mu(y)=\exp \int Q(y) \mathrm{d} y \]