By Peter V. O'Neil

ISBN-10: 0495082376

ISBN-13: 9780495082378

ISBN-10: 0935337865

ISBN-13: 9780935337860

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**Extra info for Advanced Engineering Mathematics**

**Sample text**

Therefore, v(t) = A /2gh(t) . 2 Separable Equations 19 FIGURE 1 . 9 Putting these two equations together yields dV = -kA,f2gh(t) . 5) to a specific case to illustrate its use . 9 . The tank has radius 18 feet, and water drains through a circular hole of radius 3 inches at the bottom . How long will it take the tank to empty ? 5) contains two unknown functions, V(t) and h(t), so one must be eliminated . Let r(t) be the radius of the surface of the fluid at time t and consider an interval of time from to to t, = to + At .

Then µ(x, y) is an integrating factor for M+Ny' = 0 if µ(x, y) 0 for all (x, y) in R, and µM+µNy' = 0 is exact on R . µ How do we find an integrating factor for M + Ny' = 0? 13) 34 CHAPTER 1 First-Order Differential Equation s in this region . This is a starting point . Depending on M and N, we may be able to determine A from this equation. 13) becomes simple enough to solve if we try A as a function o f just x or just y . 2 1 The differential equation x - xy - y ' = 0 is not exact . 13) is ax (-A) - ay (µ(x - x Y)) • 'Write this as - _ (x - xy) ay - xµ .

00012096 8 = H 573 0 means "approximately equal" (not all decimal places are listed) . 000120968 r Now suppose we have an artifact, say a piece of fossilized wood, and measurements show tha t the ratio of 14C to carbon in the sample is 37 percent of the current ratio . If we say that th e wood died at time 0, then we want to compute the time T it would take for one gram of th e radioactive carbon to decay this amount . 000120968 8 ' 21 9 years . This is a little less than one and one-half half-lives, a reasonable estimate if nearly s of the 14C has decayed .

### Advanced Engineering Mathematics by Peter V. O'Neil

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