Question

Solving this Differential Equation: (x^2)(d^2y/dx^2) + x(dy/dx) = 0?

Answer 1

(x^2)(d^2y/dx^2) + x(dy/dx) = 0 

Using the reduction order 

let dy/dx = w and d^2y/dx^2 = w' 

x^2w' + xw =0 w' + w/x = 0 

P(x) = 1/x Q(x) = 0 

IF = e^[∫dx/x] = e^[ln(x)] = x 

wx =∫ x(0)dx wx = C₁ w = C₁/x 

y' = w = y = ∫ w = ∫ C₁/xdx 

y = C₁ln(x) + C₂ 

Answer 2

the solution is in form of
$y=klog_{e} (x)$
substitute it in  the equation 
$x^{2} \frac{d^2y}{dx^2}+x \frac{dy}{dx}= x^{2} \frac{-k}{x^2}+x \frac{k}{x}=-k+k=0$