Question

If dy/dx = (1 + x)/(xy), x>0, and y=-4 when x=1, then when x=3, y= what?

Answer 1

y dy = (1+x)/x dx = (1 + 1/x) dx

Integrate both sides

Answer 2

After integrating and solving, I got approximately 4.711393061. Does this seem right?

Answer 3

It's -4.711393061, to find out what sign you need, you need to consider the sign of dy/dx. You can see that because y = -4 at x = 1, dy/dx is negative, so at that point y(x) is negative and decreasing. It can't become positive, because that would require there to be a point where dy/dx = 0 between x = 1 and x = 3( assuming y is continuous), but x + 1/x doesn't become zero on that interval.