Answer to Question #314505 in Differential Equations for Jyo

Question

Find the general solution of the equation:

𝑥^2𝑦" − 7𝑥𝑦′ + 12𝑦 = 0

Answer 1

$x^2y''-7xy'+12y=0\\y=x^pz:\\y'=px^{p-1}z+x^pz'\\y''=p\left( p-1 \right) x^{p-2}z+2px^{p-1}z'+x^pz''\\p\left( p-1 \right) x^pz+2px^{p+1}z'+x^{p+2}z''-7px^pz-7x^{p+1}z'+12x^pz=0\\x^{p+2}z''+\left( 2p-7 \right) x^{p+1}z'+\left( p^2-8p+12 \right) x^pz=0\\p^2-8p+12=0\Rightarrow p\in \left\{ 2,6 \right\} \\p=2:\\xz''-3z'=0\Rightarrow z'=Cx^3\Rightarrow z=C_1+C_2x^4\Rightarrow y=C_1x^2+C_2x^6$