This differential equation is separable, thus we only have to move things around and take integrals.
We have #(dy)/(dx) = (e^(-x))/(y)#, or we can also write
#(dy)/(dx) = (1)/(y*e^(x))#
Separable differential equations require our equation to have all #y#'s and #dy#'s on one side, and all #x#'s and #dx#'s on the other.
In this case, we can start off by multiplying both sides by #y#.
#y(dy)/(dx) = (1)/(cancel(y) e^(x)) * cancel(y)#
#y(dy)/(dx) = (1)/(e^(x))#
Moving our #dx# on the right by multiplying both sides the same way we get
#y(dy)/cancel(dx) * cancel(dx)= (1)/(e^(x)) * dx#
#y* dy = (1)/(e^(x)) dx#
# y* dy = e^(-x) dx#
This looks very familiar. In fact, we can integrate both sides now.
#int ydy = int e^(-x) dx#
#1/2 y^2 = -e^(-x) + C#
Our goal now is to get #y# by itself. In order to do this, we can move a few things around again.
Multiplying both sides by #2# yields
#cancel(2) * 1/cancel(2) y^2 = 2(-e^(-x) + C)#
#y^2 = 2(-e^(-x) + C)#
By taking the square root of both sides we get
#y = ± sqrt(2(-e^(-x) + C))#
So, the general solutions to our differential equation are
#y = sqrt(2(C-e^(-x)))#
and
#y = - sqrt(2(C-e^(-x)))#