Question #314505

Find the general solution of the equation:


𝑥^2𝑦" − 7𝑥𝑦′ + 12𝑦 = 0

1
Expert's answer
2022-03-20T06:40:58-0400

x2y7xy+12y=0y=xpz:y=pxp1z+xpzy=p(p1)xp2z+2pxp1z+xpzp(p1)xpz+2pxp+1z+xp+2z7pxpz7xp+1z+12xpz=0xp+2z+(2p7)xp+1z+(p28p+12)xpz=0p28p+12=0p{2,6}p=2:xz3z=0z=Cx3z=C1+C2x4y=C1x2+C2x6x^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


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