Answer to Question #119429 in Differential Equations for wayne

Question

Find the constant a so that e^xy+y=x-1 is an implicit solution to dy/dx=(e^(-xy)+ay)/(e^(-xy)+x)

Answer 1

We know that $e^{xy}+y = x-1$ or $e^{xy}+y-x+1 = 0$ . Let us take the derivative of the last expression with respect to x:

$e^{xy}\left( y+x\dfrac{dy}{dx}\right) + \dfrac{dy}{dx} - 1 + 0 = 0.$ Therefore, $\dfrac{dy}{dx} = \dfrac{1-ye^{xy}}{xe^{xy}+1} = \dfrac{e^{-xy}-y}{x+e^{-xy}}.$

We can see that $e^{-xy}-y$ should be equal to $e^{-xy}+ay$ , therefore $a=-1.$