Question

Find the solution of the differential equation that satisfies the given initial condition.

dy/dx = ln x / (xy), y(1) = 2

Answer 1

`\displaystyle(dy)/(dx) = (\ln x)/(xy), y(1)= 2.\ \int y\, dy = \int (\ln x)/(x) \, dx \ \Rightarrow\ \textstyle (1)/(2)y^2 = (1)/(2)( \ln x )^2 + C. \\ \\ \text{Now }y(1) = 2\ \Rightarrow (1)/(2)(2)^2 = (1)/(2)( \ln 1 )^2 + C\ \Righarrow\ 2 = C,\text{ so } (1)/(2)y^2 = (1)/(2) ( \ln x )^2 + 2. \\ \\ y^2 = ( \ln x )^2 + 4\ \Rightarrow\ y = \pm√( (\ln x)^2 + 4). \\ \\ \text{ Since $y(1) = 2$, we have } y = √( (\ln x)^2 + 4)`