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(a) Using the chain rule, show that = 1, and (i) dt d² (ii) = 1/²-1/dr. (iii) Hence show that (*) becomes a constant coefficient equation. (b) Use the method in (i) to find the general solution of the following equation, for t > 0: +²²+2t - 8x = t

(a) Using the chain rule, show that = 1, and (i) dt d² (ii) = 1/²-1/dr. (iii) Hence show that (*) becomes a constant coefficient equation. (b) Use the method in (i) to find the general solution of the following equation, for t > 0: +²²+2t - 8x = t

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 1CR
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4.4

4 One type of equation that occasionally occurs in economics is Euler's differential equation ²²+at+bx = 0, where t > 0, and a and b are constants. This is linear, but the coefficients are not constants. It can be solved by transforming it into one with constant coefficients by using the change of variable t = es (or, equivalently, s = lnt).
Transcribed Image Text:4 One type of equation that occasionally occurs in economics is Euler's differential equation ²²+at+bx = 0, where t > 0, and a and b are constants. This is linear, but the coefficients are not constants. It can be solved by transforming it into one with constant coefficients by using the change of variable t = es (or, equivalently, s = lnt).
(a) Using the chain rule, show that dx (i) 1 dx and dt t ds' d² dt2 d²a dx (ii) = -1 1 dr. ds² ds (iii) Hence show that (*) becomes a constant coefficient equation. (b) Use the method in (i) to find the general solution of the following equation, for t > 0: +²²+2t - 8x = t
Transcribed Image Text:(a) Using the chain rule, show that dx (i) 1 dx and dt t ds' d² dt2 d²a dx (ii) = -1 1 dr. ds² ds (iii) Hence show that (*) becomes a constant coefficient equation. (b) Use the method in (i) to find the general solution of the following equation, for t > 0: +²²+2t - 8x = t
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