anti differential help

[aff]

the differential equation is dy/dx=(1+y)/x, find the particular solution for y=f(x), with the initial condition f(-1)=1



i separated the variable and found the antiderivative and i got


ln(1+y)=1n(x)+C


what do i do after this, i'm not suppose to use a calculator?

 

[HallsofIvy]

The first thing you need to do is correct your integral! The anti-derivative of 1/x is NOT "ln(x)", it is "ln(|x|)". That's particularly important here where you are given a value of y at x=-1.

Of course, the next thing you might want to do really makes that uneccesary- solve for y. The "inverse" to the ln function is the exponential: \(\displaystyle e^{ln(x)}= x\). So to solve for y, the exponential of both sides:
\(\displaystyle e^{ln(1+ y)}= e^{ln|x|+ C}\)
\(\displaystyle 1+ y= |x|e^C\)
(of course, \(\displaystyle e^{a+ b}= e^ae^b\))
Not set x= -1 and y= 1 to determine what C should be.