What is a solution to the differential equation dy/dx=e^-x/y?

1 Answer
Jul 16, 2016

y = sqrt(2(C-e^(-x))) and y = - sqrt(2(C-e^(-x)))

Explanation:

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)))