Question

"Consider equation
x^2y''-2xy'+2y=0, x>0
What differential equation does the Wronskian W(x)=W[y1,y2](x) for any pair of solutions of the above equation satisfy?"

Answer 1

Abel's theorem states that if \(y_1(x)\) and \(y_2(x)\) are two solutions to\[y''+p(x)y'+q(x)y=0,\]then the Wronskian of the two solutions is\[W(y_1,y_2)(x)=c\exp\left[-\int p(x)\,dx\right].\]For this case, we have that \(p(x)=-2/x\). Hence\[W(y_1,y_2)(x)=cx^2.\]