## Differential Equations and Linear Algebra (4th Edition)

$y=\ln x$
Multiply the entire expression by $\frac{dx}{y}$ to separate variables. $$\frac{dx}{y}(\frac{dy}{dx}=\frac{y}{xlnx})$$ $$\frac{dy}{y}=\frac{dx}{xlnx}$$ Since each side of the equation is in terms of one variable, you can integrate. $$\int \frac{dy}{y}=\int \frac{dx}{xlnx}$$ Use a u-sub of $u=lnx$ and $du=dx/x$ for the second integral. $$\int \frac{dy}{y}=\int \frac{du}{u}$$ $$ln|y|=ln|u|$$ $$ln|y|=ln|lnx|$$ $$y=\ln x$$