Final answer:
To solve the differential equation x^2y'' - 20y = 0, assume the solution is in the form of y = x^r, substitute it into the equation, simplify it, and solve for r. The general solution is y = c1x^5 + c2x^-4.
Step-by-step explanation:
To solve the given differential equation x^2y'' - 20y = 0, we can assume that the solution is in the form of y = x^r, where r is a constant.
Substituting this into the differential equation, we get r(r-1)x^r-2 - 20x^r = 0.
Simplifying the equation, we get r^2 - r - 20 = 0.
Using the quadratic formula, we find that r = 5 or r = -4.
Therefore, the general solution to the differential equation is y = c1x^5 + c2x^-4, where c1 and c2 are constants.