workshop7

# workshop7 - between the object and the constant room...

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Workshop 7 1) a) dy dx = 2 x + 3 y , b) dy dx = e 2 x +3 y , c) dy dx = x 3 y 2 , d) dy dx = x 2 + y 3 Two of these are separable. For each of these two separable equations, solve the initial value problem with the initial condition y (0) = 1. In each case your solution should be written as y = f ( x ) where f ( x ) is a formula. Choose one of the non-separable equations and explain carefully why it is not separable. 2) A small object of unknown temperature was placed in a large room that had the ﬁxed temperature 30 C. After 10 minutes, the object’s tempera- ture is - 10 C, and after an additional 10 minutes, the object’s temperature was - 5 C. What was the initial temperature of the object? (Assume that the temperature obeys Newton’s law of cooling: the rate of change of the temperature of the object is proportional to the diﬀerence in temperature
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Unformatted text preview: between the object and the constant room temperature.) 3) Find the general solution of the diﬀerential equation e-y dy dt + 2 cos t = 0 . a) Sketch the solutions corresponding to several values of the constant of integration, C . Does every value of the constant of integration correspond to a solution curve? If not, which C ’s do occur? b) Do all the solutions have the same domain? Explain. c) Sketch the slope ﬁeld associated with this equation. Why is it not a good idea to use isoclines? Instead, sketch the slope at several points along the line t=0, and then at the corresponding points along the lines t = π/ 6, π/ 3, π/ 2, etc. Finally, superimpose your sketch of the solution curves onto the slope ﬁeld....
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