B prove that if is a solution to the differential

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Differential Equations with Boundary-Value Problems
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Chapter 12 / Exercise 24
Differential Equations with Boundary-Value Problems
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(b) Prove that if is a solution to the differential equation on ( , ) and where then for [ Hint: Consider the sign of .] (c ) Now let be a solution to the IVP (7) on ( ). Of course If vanishes at some point then let b be the largest of such points; otherwise, set . Similarly, if g vanishes at some point then let a be the smallest (furthest to the left) of such points; otherwise, set . Here we allow and . (Because g is a continuous function, it can be proved that there always exist such largest and smallest points.) Using the results of parts (a) and (b) prove that if both a and b are finite, then g has the following form: What is the form of g if ? If ? If both and ? (d) Verify directly that the above concatenated function g is indeed a solution to the IVP (7) for all choices of a and b with Also sketch the graph of several of the solu- tion function g in part (c) for various values of a and b, including infinite values. We have analyzed here a first-order IVP that not only fails to have a unique solution but has a solution set consisting of a doubly infinite family of functions (with a and b as the two parameters). Utility Functions and Risk Aversion Courtesy of James E. Foster, George Washington University Would you rather have $5 with certainty or a gamble involving a 50% chance of receiving $1 and a 50% chance of receiving $11? The gamble has a higher expected value ($6); however, it also has a greater level of risk. Economists model the behavior of consumers or other agents facing a 2 b . a q b q a q b q g ( x ) ( x a ) 3 if x a 0 if a 6 x b . ( x b ) 3 if x 7 b a q b q a 2 x 6 2, b 2 x 7 2, g g (2) 0. q , q y g ( x ) f ¿ a x b . f ( x ) 0 a 6 b , f ( a ) f ( b ) 0, q q dy / dx 3 y 2/3 y f ( x ) f ( x ) ( x c ) 3 f ( x ) dy / dx 3 y 2/3 y f ( x ) a 2 b , x 7 b , ( x b ) 3 a x b , x 6 a , ( x a ) 3 q , q dy dx 3 y 2/3 , y A 2 B 0 , Group Projects for Chapter 2 85 G H
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Differential Equations with Boundary-Value Problems
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Chapter 12 / Exercise 24
Differential Equations with Boundary-Value Problems
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risky decisions with the help of a (von Neumann–Morgenstern) utility function u and the criterion of expected utility. Rather than using expected values of the dollar payoffs, the payoffs are first transformed into utility levels and then weighted by probabilities to obtain expected utility. Following the sugges- tion of Daniel Bernoulli, we might set and then compare to [0.5 ln 1 0.5 ln 11] 0.1969, which would result in the sure thing being chosen in this case rather than the gamble. This utility function is strictly concave, which corresponds to the agent being risk averse, or wanting to avoid gambles (unless of course the extra risk is sufficiently compensated by a high enough increase in the mean or expected payoff). Alternatively, the utility function might be , which is strictly convex and corre- sponds to the agent being risk loving. This agent would surely select the above gamble. The case of occurs when the agent is risk neutral and would select according to the expected value of the payoff. It is normally assumed that at all payoff levels, x; in other words, higher payoffs are desirable.

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