Separable differential equation dy/dx=-ylny/x with y(1)=1

[dooogle]

Homework Statement

dy/dx=-ylny/x initial conditions y(x=1)=1

express answer in the form f(x,y)=0

Homework Equations

The Attempt at a Solution

i let u=ln y

du/dy=1/y

y*du=dy

subbing into equation gives int(y/yu du)=-int dx

which is equal to int(1/u du)=-int(1/x) dx

ln (u)=-ln (x) +c

subbing back in for u gives ln(ln (y))=-ln (x)+c

ln (y)=1/x +c

when x= 1 y=1

ln(1)=1/1 +c

0=1+c

c=-1

equation becomes ln(ln(y))=-ln(x)+c

rearranging (here I am concerned that i make a mistake but I am not sure)

ln(ln(y))+ln(x)+1=0

thank you for your time

dooogle

 

[vela]

It looks like you used c to mean two different things near the end. You need to fix that.

You could solve for y explicitly in terms of x and plug that back into the original equation to verify it's a solution.