Advanced Engineering Mathematics, 6th Edition by Peter V. O'Neil PDF

By Peter V. O'Neil

ISBN-10: 0495082376

ISBN-13: 9780495082378

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Y = 3x2 y + 2 y 2 = 8 1. 3y = 4x/y2 13. ln yx y = 3x2 y y 2 = e3 2. y + xy = 0 14. 2yy = ex−y y 4 = −2 3. cos y y = sin x + y 15. yy = 2x sec 3y y 2/3 = /3 4. e x+y y = 3x 5. xy + y = y 6. y = 2 x + 1 2 − 2y 2y 7. x sin y y = cos y 8. 2y2 + 1 x y = y x+1 9. y + y = ex − sin y 10. cos x + y + sin x − y y = cos 2x 2 16. An object having a temperature of 90 degrees Fahrenheit is placed into an environment kept at 60 degrees. Ten minutes later the object has cooled to 88 degrees. What will be the temperature of the object after it has been in this environment for 20 minutes?

This is called the initial mass. Then m 0 =A=M so m t = Mekt If at some later time T we find that there are MT grams, then m T = MT = MekT Then ln MT M k= 1 MT ln T M = kT hence This gives us k and determines the mass at any time: m t = Meln MT /M t/T We obtain a more convenient formula for the mass if we choose the time of the second measurement more carefully. Suppose we make the second measurement at that time T = H at which exactly half of the mass has radiated away. At this time, half of the mass remains, so MT = M/2 and MT /M = 1/2.

The strategy is to find the general solution of this linear equation and from it produce the general solution of the original Riccati equation. 28 Consider the Riccati equation 1 1 2 y = y2 + y − x x x By inspection, y = S x = 1 is one solution. Define a new variable z by putting y = 1+ 1 z Then y =− 1 z z2 Substitute these into the Riccati equation to get − 1 1 1 z = 1+ 2 z x z 2 + 1 1 2 1+ − x z x or 3 1 z + z=− x x This is linear. An integrating factor is e 3/x dx = x3 . Multiply by x3 to get x3 z + 3x2 z = x3 z = −x2 Integrate to get 1 x3 z = − x3 + C 3 so 1 C z x =− + 3 3 x The general solution of the Riccati equation is y x = 1+ 1 1 = 1+ zx −1/3 + C/x3 This solution can also be written yx = in which K = 3C is an arbitrary constant.

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Advanced Engineering Mathematics, 6th Edition by Peter V. O'Neil

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