Chapter 1 - First-Order Differential Equations - 1.9 Exact Differential Equations - True-False Review - Page 91: i

False

$(e^{x\sin y}\sin y)dx+(e^{x\sin y}\cos y)dy=0$ __(1) Here, $M(x,y)=e^{x\sin y}\sin y$ $N(x,y)=e^{x\sin y}\cos y$ $M_y=x\cos y\;e^{x\sin y}\sin y+e^{x\sin y}\cos y$ $\Rightarrow M_y=e^{x\sin y}\cos y[x\sin y+1]$ $N_x=\sin y\;e^{x\sin y}\cos y$ $\Rightarrow M_y\neq N_x$ That's why (1) is not exact differential equation.