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Unformatted text preview: Diﬀerential Equations Practice
1. Suppose that y (x) is a function which is never zero and satisﬁes the diﬀerential equation −y = 2xy 2 . Suppose also that y (0) = 3. Find y . 2. Is it possible that y 2 + (y )2 = 1? If not, explain why. If so, give an example. 3. Revolve the region in the ﬁrst quadrant bounded by the graph of the function y = −x2 + h, h > 0, about the y -axis. The 3-dimensional solid obtained is the shape of a melting heap of snow when it is of height h. (a) Express the volume V of the heap of snow as a function of its height h. (b) The heap of snow melts at a rate equal to the area of its base. What is the rate at which its height is decreasing? (Hint: ‘Rate’ means derivative. Solve for dh using what you know about dV .) dt dt (c) Assuming that h = 60 when t = 0, what is the rate of melting when t = 50? 4. The third-life of a radioactive element is deﬁned to be the time required for two thirds of the radioactive nuclei present in a sample to decay (leaving one third remaining). (a) Prove that the third-life is a constant that does not depend on the number of radioactive nuclei intially present in the sample. (b) Suppose you are given a substance with a third-life of one month. What is its half-life? 5. A cup of hot cocoa (initial temperature: 90 ◦ C) is left outside on a winter day for 10 minutes, during which time it cools to a lukewarm 40 ◦ C. It is then taken back inside the house where it cools further to a tepid 25 ◦ C after another 20 minutes. The ambient temperature inside the house is 20 ◦ C. Assuming that the k -value (in Newton’s law of cooling, section 7.2) is a property of the cocoa and therefore is constant throughout this scenario, how cold is it outside? ...
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