Solve the equation dy/dx=1/x*((lnx^2)-1).

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to separate the variables, hence you should multiply by `dx`  both sides such that:

`dy = (dx)/(x*((lnx^2)-1))`

You need to integrate both sides to find the general solution to differential equation such that:

`y = int (dx)/(x*((lnx^2)-1)`

You should come up with the substitution `ln x = t` , hence, differentiating both sides yields `(dx)/x = dt`

Hence, you need to substitute t for ln x such that:

`y = int (dt)/(2t-1)`

You need to come up with the next substitution 2t-1=u, hence, differentiating both sides yields `2dt = du =gt dt = (du)/2` .

Hence, you need to substitute u for t such that:

`y = int ((du)/2)/(u) =gt y = (1/2)*ln|u| + c`

You need to substitute `2t - 1`  for u such that:

`y = (1/2)*ln|2t - 1| + c`

You need to substitute `ln x`  for t such that:

`y = (1/2)*ln|2ln x- 1| + c`

`y = (1/2)*ln|ln x^2- 1| + c`

Hence, evaluating the general solution to equation yields `y = (1/2)*ln|ln x^2 - 1| + c.`

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