Find the general solution of the equation:
𝑥^2𝑦" − 7𝑥𝑦′ + 12𝑦 = 0
x2y′′−7xy′+12y=0y=xpz:y′=pxp−1z+xpz′y′′=p(p−1)xp−2z+2pxp−1z′+xpz′′p(p−1)xpz+2pxp+1z′+xp+2z′′−7pxpz−7xp+1z′+12xpz=0xp+2z′′+(2p−7)xp+1z′+(p2−8p+12)xpz=0p2−8p+12=0⇒p∈{2,6}p=2:xz′′−3z′=0⇒z′=Cx3⇒z=C1+C2x4⇒y=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^6x2y′′−7xy′+12y=0y=xpz:y′=pxp−1z+xpz′y′′=p(p−1)xp−2z+2pxp−1z′+xpz′′p(p−1)xpz+2pxp+1z′+xp+2z′′−7pxpz−7xp+1z′+12xpz=0xp+2z′′+(2p−7)xp+1z′+(p2−8p+12)xpz=0p2−8p+12=0⇒p∈{2,6}p=2:xz′′−3z′=0⇒z′=Cx3⇒z=C1+C2x4⇒y=C1x2+C2x6
Need a fast expert's response?
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS!
Comments