Question

It is easy to check that for any value of x=y*e^(C*y+1), the function is solution of equation y'=y/(x(ln(x)-ln(y)). Find the value of for which the solution satisfies the initial condition y(1)=1 . c=?

Answer 1

at x=1 y=1
\[x=y*e ^{Cy+1}\]
\[1=1*e ^{C.1+1}\]
\[1=e ^{C+1}\]
\[ln(1)=ln(e ^{C+1})\]
\[0=C+1\]
\[C=-1\]