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Huu hapa ni uthibitisho: Tunapewa y = ( x). Tunahitaji kuthibitisha kwamba x^2 (d^2y)/(dx^2) + x (dy)/(dx) + y = 0. Step 1: Tafuta derivative ya kwanza, (dy)/(dx). Tunatumia chain rule. y = ( x) (dy)/(dx) = ( x) · (d)/(dx)( x) Tunajua kwamba (d)/(dx)( x) = (1)/(x). (dy)/(dx) = ( x) · (1)/(x) (dy)/(dx) = (( x))/(x) Step 2: Tafuta derivative ya pili, (d^2y)/(dx^2). Tunatumia quotient rule kwa (dy)/(dx) = (( x))/(x). Kumbuka: (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2). Hapa, u = ( x) na v = x. u' = (d)/(dx)(( x)) = -( x) · (1)/(x) = -(( x))/(x). v' = (d)/(dx)(x) = 1. Sasa badilisha hizi kwenye fomula ya quotient rule: (d^2y)/(dx^2) = ((-( x))/(x)) · x - ( x) · 1x^2 (d^2y)/(dx^2) = (-( x) - ( x))/(x^2) Step 3: Badilisha y, (dy)/(dx), na (d^2y)/(dx^2) kwenye equation x^2 (d^2y)/(dx^2) + x (dy)/(dx) + y = 0. Chukua upande wa kushoto wa equation (LHS): x^2 ((-( x) - ( x))/(x^2)) + x ((( x))/(x)) + ( x) Rahisisha maneno: (-( x) - ( x)) + (( x)) + ( x) Panga upya maneno: -( x) + ( x) - ( x) + ( x) 0 + 0 = 0 Kwa kuwa upande wa kushoto ni sawa na upande wa kulia (RHS = 0), tumethibitisha equation. That's 2 down. 3 left today — send the next one.