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

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

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

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

2 a x^{2} \frac{d y}{d x}^{2} + 2 a x^{2} y \frac{d ^{2} y}{d x ^{2}} + 8 a x y \frac{d y}{d x} + 2 b x \frac{d y}{d x}^{2} + 2 b x y \frac{d ^{2} y}{d x ^{2}} + 2 c e^{- 3 x} x \frac{d y}{d x}^{2} + 2 c e^{- 3 x} x y \frac{d ^{2} y}{d x ^{2}} - 12 c e^{- 3 x} x y \frac{d y}{d x} + 9 c e^{- 3 x} x y^{2} + 2 a y (x)^{2} + 4 b y (x) \frac{d y}{d x} + 4 c e^{- 3 x} y \frac{d y}{d x} - 6 c e^{- 3 x} y^{2}

Solution steps

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

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

= \frac{d}{d x} (2 y \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^{2})

\frac{d}{d x} (2 y \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^{2}) = 2 a x^{2} \frac{d y}{d x}^{2} + 2 a x^{2} y \frac{d ^{2} y}{d x ^{2}} + 8 a x y \frac{d y}{d x} + 2 b x \frac{d y}{d x}^{2} + 2 b x y \frac{d ^{2} y}{d x ^{2}} + 2 c e^{- 3 x} x \frac{d y}{d x}^{2} + 2 c e^{- 3 x} x y \frac{d ^{2} y}{d x ^{2}} - 12 c e^{- 3 x} x y \frac{d y}{d x} + 9 c e^{- 3 x} x y^{2} + 2 a y (x)^{2} + 4 b y (x) \frac{d y}{d x} + 4 c e^{- 3 x} y \frac{d y}{d x} - 6 c e^{- 3 x} y^{2}

= 2 a x^{2} \frac{d y}{d x}^{2} + 2 a x^{2} y \frac{d ^{2} y}{d x ^{2}} + 8 a x y \frac{d y}{d x} + 2 b x \frac{d y}{d x}^{2} + 2 b x y \frac{d ^{2} y}{d x ^{2}} + 2 c e^{- 3 x} x \frac{d y}{d x}^{2} + 2 c e^{- 3 x} x y \frac{d ^{2} y}{d x ^{2}} - 12 c e^{- 3 x} x y \frac{d y}{d x} + 9 c e^{- 3 x} x y^{2} + 2 a y (x)^{2} + 4 b y (x) \frac{d y}{d x} + 4 c e^{- 3 x} y \frac{d y}{d x} - 6 c e^{- 3 x} y^{2}

(x^{4} + y^{4}) d x - x y^{3} d y

sin (a) (1 + tan (a)) + cos (a) (1 + cot (a))

(x + \frac{2}{3})^{5}

(y^{2} - 2 x y) d x + x^{2} d y

(1 + cos^{2} (x)) (cos (x))

What is (d^2)/(dx^2)((y^2)(ax^2+bx+cxe^{-3x})) ?
The answer to (d^2)/(dx^2)((y^2)(ax^2+bx+cxe^{-3x})) is