(d^2)/(dx^2)((y)(ax^2+bx+cxe^{-3x}))

解答

\frac{d ^{2}}{d x ^{2}} ((y) (a x^{2} + b x + c x e^{- 3 x}))

\frac{d ^{2} y}{d x ^{2}} (a x^{2} + b x + c e^{- 3 x} x) + 2 (2 a x + b + c (e^{- 3 x} - 3 e^{- 3 x} x)) \frac{d y}{d x} + (2 a + c (9 e^{- 3 x} x - 6 e^{- 3 x})) y

求解步骤

\frac{d ^{2}}{d x ^{2}} ((y) (a x^{2} + b x + c x e^{- 3 x}))

\frac{d}{d x} (y (a x^{2} + b x + c x e^{- 3 x})) = \frac{d y}{d x} (a x^{2} + b x + c x e^{- 3 x}) + (2 a x + b + c (e^{- 3 x} - 3 e^{- 3 x} x)) y

= \frac{d}{d x} (\frac{d y}{d x} (a x^{2} + b x + c x e^{- 3 x}) + (2 a x + b + c (e^{- 3 x} - 3 e^{- 3 x} x)) y)

\frac{d}{d x} (\frac{d y}{d x} (a x^{2} + b x + c x e^{- 3 x}) + (2 a x + b + c (e^{- 3 x} - 3 e^{- 3 x} x)) y) = \frac{d ^{2} y}{d x ^{2}} (a x^{2} + b x + c e^{- 3 x} x) + 2 (2 a x + b + c (e^{- 3 x} - 3 e^{- 3 x} x)) \frac{d y}{d x} + (2 a + c (9 e^{- 3 x} x - 6 e^{- 3 x})) y

= \frac{d ^{2} y}{d x ^{2}} (a x^{2} + b x + c e^{- 3 x} x) + 2 (2 a x + b + c (e^{- 3 x} - 3 e^{- 3 x} x)) \frac{d y}{d x} + (2 a + c (9 e^{- 3 x} x - 6 e^{- 3 x})) y

e x p an d \frac{( x - 2 ) ( x - 4 )}{3}

e x p an d \frac{2 ( 6 - x ^{2} ) ^{2}}{x ^{2}}

e x p an d 2 π - 2 π x (2 - 2 x^{2})

e x p an d \frac{2 - 2 x ^{2}}{( x ^{2} + 1 ) ^{2}}

e x p an d (\frac{1}{3 x} + (- 3) x^{2})^{8}