Advanced Engineering Mathematics by Peter V. O'Neil

By Peter V. O'Neil

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This is a little less than one and one-half half-lives, a reasonable estimate if nearly s of the 14C has decayed . 1 3 (Torricelli's Law) Suppose we want to estimate how long it will take for a container to empt y by discharging fluid through a drain hole . This is a simple enough problem for, say, a sod a can, but not quite so easy for a large oil storage tank or chemical facility . We need two principles from physics . The first is that the rate of discharge of a fluid flowing through an opening at the bottom of a container is given b y dV = -kAv , dt in which V(t) is the volume of fluid in the container at time t, v(t) is the discharge velocit y of fluid through the opening, A is the cross sectional area of the opening (assumed constant) , and k is a constant determined by the viscosity of the fluid, the shape of the opening, and th e fact that the cross-sectional area of fluid pouring out of the opening is slightly less than tha t of the opening itself.

8) . To verify this, differentiate the last equation implicitly with respect to x : 2xy 3 + 3x2 y2 y ' + 2 + 8e4j'y' = 0 , or 2xy3 + 2 + (3x2 y2 + 8e4j' )y ' = 0. This is equivalent to the original differential equation Y' 2xy3 + 2 3x 2y2 +8e4y With this as background, we will make the following definitions . 4 M(x, y) and a ay= * N( x, Y) Exact Differential Equatio n When a potential function exists on a region R for the differential equation then this equation is said to be exact on R . 17 is exact (over the entire plane), because w e exhibited a potential function for it, defined for all (x, y) .

2 Separable Equations 13 Er: EXAMPLE 1 . 8 x2y ' = 1 +y is separable, and we can writ e 1+Y dy = x2 dx . The algebra of separation has required that x 0 and y # -1, even though we can put x = 0 and y = -1 into the differential equation to obtain the correct equation 0 = O . Now integrate the separated equation to obtai n 1n11+YJ 1 =--1 +k. This implicitly defines the general solution . In this case, we can solve for y(x) explicitly . Begi n by taking the exponential of both sides to obtai n -'/x 11-f -y* = e k e = Ae -'fix , in which we have written A = e k .

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