Engineering Board Exam Reviewer · Second-Order Linear Homogeneous Equations · hard

Find the general solution of x2y''-2xy'+2y=0 for x>0.

1. y = C1 x + C2x2
2. y = C1 x2 + C2 x3
3. y = C1 + C2 x2
4. y = C1 x + C2 x2

Answer

D. y = C1 x + C2 x2

Solution

This is a Cauchy-Euler equation. Let y=xm: m(m-1)-2m+2=0 m2-3m+2=0 m=1,2. Hence y=C1x+C2x2.

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