By Peter V. O'Neil

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14. 15. 16. ln ( yx)y' = 3x2y; y(2) = e3 2yy' = ex-Y2 ; y(4) = - 2 yy' = 2x sec(3y) ; y(2/3) = 7r/3 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 ha s been in this environment for 20 minutes? How lon g will it take for the object to cool to 65 degrees ? 17. A thermometer is carried outside a house whose ambient temperature is 70 degrees Fahrenheit .

This equation forces us to choose h so that h ' (x) = 2, and we may therefore set h(x) = 2x. This gives co(x , y) = xZy3 + 2e 4y + 2x, as we got before . Not every first-order differential equation is exact . For example, conside r Y + y' _ 30 CHAPTER 1 First-Order Differential Equation s If there were a potential function cp, then we would hav e a* - y' ay = 1, Integrate acp/ax = y with respect to x to get cp(x, y) = xy + g(y) . Substitute this into &p/ay = 1 to get c3y (xy + g( y)) = x+g (y) = 1 .

Terminal Velocity Consider an object that is falling under the influence of gravity, in a medium such as water, air or oil . This medium retards the downward motion of the object. Think, for example, of a brick dropped in a swimming pool or a ball bearing dropped in a tan k of oil . We, want to analyze the object's motion . _ Let v(t) be the velocity at time t . The force of gravity pulls the object down and ha s magnitude nag . The medium retards the motion. The magnitude of this retarding force is not obvious, but experiment has shown that its magnitude is proportional to the square o f the velocity .

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