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4. Consider the difference equation N(t + 1) = 0.5N(t) + 40 where we are given that N(0) = 100. a) Check that N(t) = 100* (0.5) is not a solution to this difference equation. b) Check that N(t) = 100* (0.5) +40 is not a solution to this difference equation. c) Check that N(t) = 20 (0.5) +80 IS a solution to this difference equation. (You might be wondering where this came from. You will verify a general formula for solu- tions to linear difference equations with constant terms in the next problem.)

4. Consider the difference equation N(t + 1) = 0.5N(t) + 40 where we are given that N(0) = 100. a) Check that N(t) = 100* (0.5) is not a solution to this difference equation. b) Check that N(t) = 100* (0.5) +40 is not a solution to this difference equation. c) Check that N(t) = 20 (0.5) +80 IS a solution to this difference equation. (You might be wondering where this came from. You will verify a general formula for solu- tions to linear difference equations with constant terms in the next problem.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Thus far in the class we've only seen one kind of so-called "linear" difference equation models: the Malthusian ones of the form N(t+1)= XN (t) This is called linear because N(t+1) is a linear function of N(t) - in other words, if we view N(t+1) as "y" and N(t) as "r" then this is an equation of the form "y = mx" since À is a constant. Another more general type of "linear" difference equation is: N(t+1) = AN(t) + a where a is some constant. This is again a linear model because N(t+1) is a linear function of N(t), this time of the form "y=mx+b". This is sometimes called a "linear difference equation with a constant term". The next few problems have to do with linear difference equations with constant terms. 4. Consider the difference equation N(t + 1) = 0.5N(t) + 40 where we are given that N(0) = 100. a) Check that N(t) = 100* (0.5) is not a solution to this difference equation. b) Check that N(t) = 100* (0.5) +40 is not a solution to this difference equation. c) Check that N(t) = 20 * (0.5) + 80 IS a solution to this difference equation. (You might be wondering where this came from. You will verify a general formula for solu- tions to linear difference equations with constant terms in the next problem.) 5. Verify that N(t) = crt + is a solution to the difference equation N(t+1) = rN(t)+a. The value of c will depend on what N(0) is, but unlike in our Malthusian model, it is not as nice as simply being N(0). (Where did this come from? Messing around with the equation and making some educated guesses.) a 1-r 6. Suppose that five fish are added to an aquarium every month, 90% of those present survive every month, and there is no reproduction. a) Write down a difference equation that describes the size of the fish population in this scenario. Specify what the time step is.
Transcribed Image Text:Thus far in the class we've only seen one kind of so-called "linear" difference equation models: the Malthusian ones of the form N(t+1)= XN (t) This is called linear because N(t+1) is a linear function of N(t) - in other words, if we view N(t+1) as "y" and N(t) as "r" then this is an equation of the form "y = mx" since À is a constant. Another more general type of "linear" difference equation is: N(t+1) = AN(t) + a where a is some constant. This is again a linear model because N(t+1) is a linear function of N(t), this time of the form "y=mx+b". This is sometimes called a "linear difference equation with a constant term". The next few problems have to do with linear difference equations with constant terms. 4. Consider the difference equation N(t + 1) = 0.5N(t) + 40 where we are given that N(0) = 100. a) Check that N(t) = 100* (0.5) is not a solution to this difference equation. b) Check that N(t) = 100* (0.5) +40 is not a solution to this difference equation. c) Check that N(t) = 20 * (0.5) + 80 IS a solution to this difference equation. (You might be wondering where this came from. You will verify a general formula for solu- tions to linear difference equations with constant terms in the next problem.) 5. Verify that N(t) = crt + is a solution to the difference equation N(t+1) = rN(t)+a. The value of c will depend on what N(0) is, but unlike in our Malthusian model, it is not as nice as simply being N(0). (Where did this come from? Messing around with the equation and making some educated guesses.) a 1-r 6. Suppose that five fish are added to an aquarium every month, 90% of those present survive every month, and there is no reproduction. a) Write down a difference equation that describes the size of the fish population in this scenario. Specify what the time step is.
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