Question

If $\frac{dy}{dx}=e^{-2y}$ and y=0 when x=5, then find the value of x when y=3.

Answer 1

Given that, $\frac{dy}{dx}=e^{-2y}\Rightarrow \frac{dy}{e^{-2y}}=dx$
$\Rightarrow \int e^{2y}dy=\int dx\Rightarrow \frac{e^{2y}}{2}=x+C$ ...(i)
when x=5 and y=0, then substituting these values in Eq. (i), we get
$\frac{e^0}{2}=5+C\Rightarrow \frac{1}{2}=5+C\Rightarrow C=\frac{1}{2}-5=-\frac{9}{2}$
Eq. (i) becomes $e^{2y}=2x-9$
When y=3, then $e^6=2x-9\Rightarrow 2x=e^6+9\therefore x=\frac{(e^6+9)}{2}$