Separation of variables Calculus The given differential equation I need to solve is $dy/dx=1/x$ with the initial conditon of $x=1$ and $y=10$
My attempt: 
$dy=\frac 1x dx$
Integrating yields 
\begin{align*} 
   y&=\log x+C\\
   x=1 & y=10\\
 10 &=\log 1 + C\\
  10=C\\
\leadsto y&=\log x+10
\end{align*}
Something seems off with my work, if anyone could give me a clue where I went wrong. Thanks

Your work is fine.  Good job. You might consider an alternative solution: $$\frac{{\rm d}y}{{\rm d}x} = \frac{1}{x} \implies \int \frac{{\rm d}y}{{\rm d}x} \,{\rm d}x = \int \frac{1}{x}\,{\rm d}x \implies y = \ln x + C, $$and now we proceed as you did to get $C = 10 $. My point is: it works, but what does it mean to multiply thorugh ${\rm d}x$, etc?
