This equation is actually a common form of differential equation, called a Cauchy-Euler equation.
Here, though, the differential equation is treated as an operator on a function. In order to find the result, we need to figure some things out.
We are given:
` y (x) = ln (x)`
We now need to calculate the first and second derivative to fit the operator's action on the function. We then know these derivatives are as follows:
`y' (x) = 1 / x`
`y'' (x) = -1 / x^2`
Now, to compute `L(ln(x))` we just substitute the values we have above for y, its first derivative, and its second derivative:
`L(ln(x)) = ax^2*-1/x^2 + bx*1/x + clnx`
`L(ln(x)) = -ax^2/x^2 + bx/x + clnx`
Because x is strictly greater than zero, we can cancel all of the x terms in the fractions. We can now simplify:
`L(ln(x)) = -a + b +clnx`
And there is your answer. You unfortunately can't simplify much more.
If you wanted to have fun and get everything into a single logarithm you could do the following:
`-a+b+clnx = lne^-a + lne^b +lnx^c = ln(e^(b-a)x^c)`
Again, just for fun! I think the first answer is likely the best.
I hope all that helped you out!