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[fortpost]

9. xy'''+y''=x^-1/2

y''=p(x)
y'''=p'(x)

xp'+p=x^-1/2
p'+p/x=x^-3/2

p'+p/x=0
p=Ce^-∫dx/x=Ce^-ln|x|=C/x

p=C(x)/x
(C(x)/x)'+C(x)/x²=x^-3/2
C'(x)/x-C(x)/x²+C(x)/x²=x^-3/2
C'(x)=x^-1/2
C(x)=∫x^-1/2dx=2x^1/2+C₁

p=(2x^1/2+C₁)/x=2x^-1/2+C₁/x

y'=∫pdx=∫(2x^-1/2+C₁/x)dx=4x^1/2+C₁ln|x|+C₂
y=∫y'dx=∫(4x^1/2+C₁ln|x|+C₂)dx=(8/3)x^3/2+C₁(xln|x|-x)+C₂x+C₃