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

Separable, $y=\sqrt{2} \times |ln(x)|$
Since the equation can be manipulated so that only one variable appears on each side, the differential equation is seperable. To separate variables, multiply each side of the equation by $ydx$ to get $$y(dx)(\frac{dy}{dx}=\frac{2\ln x}{xy})$$ $$ydy=\frac{2\ln x}{x} dx$$ Since each side of the equation is in terms of one variable, you can integrate. $$\int ydy =\int \frac{2\ln x}{x} dx$$ For the second integral, use a u-sub of $u=\ln x$ and $du=dx/x$. $$\frac{1}{2}y^2=2\int udu$$ $$\frac{1}{2}y^2=ln(x)^2$$ Solve for $y$. $$y=\sqrt{2(\ln x)^2}=\sqrt{2} \times |ln(x)|$$