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Find constants to make a derivative equation true

Let

    \[ g(x) = (ax^2+bx+c) \sin x + (dx^2 + ex + f) \cos x. \]

Determine values for the constants a,b,c,d,e,f such that

    \[ g'(x) = x^2 \sin x. \]


We compute the derivative,

    \begin{align*}  g'(x) &= (2ax + b) \sin x + (ax^2+bx+c) \cos x + (2dx + e) \cos x - (dx^2 + ex + f) \sin x \\  &= (-dx^2 + (2a-e)x + (b+f)) \sin x + (ax^2 + (b + 2d)x + (c+e)) \cos x. \end{align*}

Since g'(x) = x^2 \sin x we must have

    \[ -dx^2 + (2a-e)x + b+f = x^2 \qquad \text{and} \qquad ax^2 + (b+2d)x + c+e = 0. \]

From the equation on the left, equating like powers of x we have,

    \[ d = -1, \quad 2a-e = 0 \quad b+f = 0. \]

From the equation on the right, equating like powers of x we have,

    \[ a = 0, \quad b+2d = 0, \quad c+e = 0. \]

Since a = 0 and 2a-e =0 we have e = 0.
Since e = 0 and c+e = 0 we have c = 0.
Since b + 2d = 0 and d = -1 we have b = 2.
Finally, since b + f = 0 and b = 2 we have f = -2.
Putting these all together,

    \[ a = 0, \quad b = 2, \quad c = 0, \quad d = -1, \quad e = 0, \quad f = -2. \]

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