# Find the solution of the differential equation `dy/dx=(ln x)/(xy)` that satisfies the given initial condition y(1)=2

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It is given that `dy/dx = (ln x)/(x*y)` and `y(1) = 2`

`dy/dx = (ln x)/(x*y)`

=> `y*dy = (ln x/x) dx`

Integrate both the sides

=> `y^2/2 = (ln x)^2/2 + C`

=> `y^2 = (ln x)^2 + C`

As `y(1) = 2`

=> `2^2 = 0 + C`

=> `C = 4`

`y^2 = (ln x)^2 + 4`

**The solution of the differential equation is **`y^2 = (ln x)^2 + 4`

`dy/dx=logx/(xy)` `ydy=(logx/x)dx` `ydy=logx(dx/x)`

`ydy=logx(dlogx)` `y^2/2=(log^2x)/2+c`

`y^2=log^2x+2c` `y=sqrt(log^2x+2c)`

since y(1)=2:

`2=sqrt(2c)` `c=2` `y=sqrt(log^2x+2)`