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Find a vector valued function satisfying given properties

If A and B are fixed vectors and F is a vector valued function with

    \[ F''(t) = tA + B, \]

find F(t) if F(0) = D and F'(0) = C.


We have

    \begin{align*}  && F''(t) &= tA + B  \\  \implies && \int F''(t) \,dt &= \int (tA+B) \, dt \\  \implies && F'(t) &= \frac{1}{2}t^2 A + tB + C \\  \implies && F(t) &= \frac{1}{6} t^3 A + \frac{1}{2}t^2 B + tC + D. \end{align*}

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