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7.
A scientist is modelling the amount of a chemical in the human bloodstream. The amount x of the chemical, measured in mg l –1, at time t hours satisfies the differential equation 2
d2 x dx 2 x 2 − 6 = x 2 − 3x 4 , x > 0. dt dt
(a) Show that the substitution y =
1 transforms this differential equation into x2
d2 y + y = 3. dt 2
I (5)
(b) Find the general solution of differential equation
I . (4)
Given that at time t = 0, x =
1 dx and = 0, 2 dt
(c) find an expression for x in terms of t, (4) (d) write down the maximum value of x as t varies. (1) ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________
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