It is understood that the metabolites in the morphine is function of body weight

Question

It is understood that the metabolites in the morphine is function of body weight (x) of the patient in kilograms. It is important to know the covariance on clearance level (y) before applying the medicine for flight (i.e. pilot, passenger, etc). The doctor forgot the original differential equations. He thinks the equation is in 2nd order. The experimental solution is given in the following form y = a X^b, where a and b are constant. Your task is to reconstruct the "original 2^nd order differential equations", meaning solve for alpha, beta, gamma in terms of a and b. alpha y" beta y' + yy = 0 y(0) = 0,y'(0) = 0 where y' = dy/dx and y" dx = y" = d^2 y/dx^2 Is the doctor memory correct or not about the original equation is 2^nd order? Solve the following ODE problem my" + cy' + ky = F(t) y(0) = 0, y'(0) = 0

Explanation / Answer

(4)

1)Assume the solution is of the type y =axb .

It is not possible to find a second order differential equation with constant coefficients with this solution .

(Reason: any such solution is of the form epx , where p is a scalar )

2) Nevertheless, it is correct to assume that y=axb satisfies a second order differential equation , as there are two constants a and b.

To formulate such an eqaution , eliminate a and b between the three equations given below:

                                                           y = axb ......................................................(1)

                                                           y' = abxb-1 .........................................(2)

                                                           y'' = ab(b-1)xb-1 ..................................(3)

From the first two equations, we obtain

                                                          y=bxy'....................................................(4)

and from (2) and (3),

                                                         xy"= (b-1)y'................................................(5)

           From (4) and (5) , we get

                                                       xy" = [y/xy' -1]y'

which is the required second order differential equation . (note that this is not a constant coefficient ODE)

                               

                                        

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