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

Textbook Question

Chapter 9, 9.3 - Problem 12 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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tiburtius | High School Teacher | (Level 2) Educator

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This is differential equation with separable variables. We can separate the variables by putting `y` on the left side and `dx` to right side to get

`int y dy=int (ln x)/x dx`                                                                 (1)

This looks like we multiplied the whole equation by `y dx` but that is not really the case. If you want to learn more about theory behind this I would suggest you read

W. E. Boyce, R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems   ``

or some other book on ordinary differential equations.

Now back to solving the equation by integrating. Let us first integrate the right side because it is harder.

`int ln x/x dx=`

We make substitution `t=ln x=>dx/x=dt`

`int t dt=t^2/2=(ln^2x)/2+C`

 Now we return to (1).

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

`y^2=ln^2x+C`

`y=pm sqrt(ln^2x+C)`

Now we use initial condition `y(1)=2.`

`2=pm sqrt(ln^2 1+C)`

`2=pm sqrt C`

Obviously there is no solution for minus sign.

`C=4`          

Therefore, the solution is `y=sqrt(ln^2x+4)` 

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