dy/dx - e^(x-y) = e^(x-y)

Question

Find:

\(\frac{dy}{dx} - e^{x-y} = e^{x-y}\)

dy/dx - e^(x-y) = e^(x-y)

Answer 1

\(\frac{dy}{dx} - e^{x-y} = e^{x-y}\)

⇒ \(\frac{dy}{dx} = 2e^{x-y}\)----(i)

Let x - y = t

Then 1 - \(\frac{dy}{dx}=\frac{dt}{dx}\) 

\(\therefore\) \(\frac{dt}{dx} = 1-2e^t\)  (From (i))

⇒ \(\frac{dt}{1-2e^t}=dx\) 

⇒ \(\frac{e^{-t}dt}{e^{-t}-2}=dx\) 

⇒ \(\int\frac{e^{-t}}{e^{-t}-2}dt = \int dx\) 

⇒ -log|e^-t - 2| = x + c

\(\therefore\) -log|e^{-(x - y)} - 2| = x + c is a solution of given differential equation.