Find the solution of the differential equation that satisfies the given initial condition. `(dy)/(dx) = ln(x)/(xy)` `y(1) = 2`
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).
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.
Therefore, the solution is `y=sqrt(ln^2x+4)`