By Charles R. MacCluer

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X. 15 as an "element" of the flow. Clearly, from everything we have said so far, the most important variables for our analysis are the concentration of waste in the blood (call this quantity p(x)) and the concentration of waste in the dialyzate (call this quantity q(x)). There is in fact a standard physical law governing the passage of waste material through the membrane. This is Fick's law. The enunciation is The amount of material passing through the membrane is proportional to the difference in concentration.

Now we resubstitute p =y' to find that y' =eY + C or dy - =eY+C. dx Because of the initial condition [dy/dx](O) = 1, we may conclude right away thatC=0. Thus our equation is dy dx =eY or dy -=dx. eY This may be integrated to -e-y =x + D. Of course we can rewrite the equation finally as y = -ln( -x + E). Since y(O) =0, we conclude that y(x) = -ln( -x is the solution of our initial value problem. • + 1) 38 Chapter 1 What Is a Differential Equation? EXERCISES 1. Solve the following differential equations using the method of reduction of order: (a) yy"+ (y')2=0 (e) 2yy"= 1 + (y')2 (b)xy"= y' + (y')3 (c)y"-k2y=0 (f) yy"- (y')2=0 (g)xy"+y'=4x (d)x2y"=2xy' + (y'f 2.

First observe that we can differentiate the given equation to obtain 2x +2y The constant c dy · - dx = 0. 5). 5 We rewrite the differential equation as dy dx x = y for the family of orthogonal trajectories. Now taking negative reciprocals, as indicated in the discussion right before this example, we obtain the new differential equation dy y dx x for the family of orthogonal trajectories. We may easily separate variables to obtain 1 -dy y 1 = -dx. x 24 Chapter 1 What Is a Differential Equation? Applying the integral to both sides yields f� dy = f� dx or In IYI = In lxl + C.