My first thought would be to go ahead and do the derivative on the left side: so that . That's an "Euler type" equation- also called and "equi-potential" equation. The change of variable t= ln(x) will convert it to a linear equation with constant coefficients.
A "short cut" that will work here is to try , , and . Putting those into the equation . In order that this be true for all x, we must have . That quadratic equation has roots . The general solution would be for constants A and B.