(a) Using the chain rule, show that da (i) da dt 1, and ds' = x (ii) = 1/²-1/2 ds2 (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: 1²+2t - 8x = t
(a) Using the chain rule, show that da (i) da dt 1, and ds' = x (ii) = 1/²-1/2 ds2 (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: 1²+2t - 8x = t
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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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
1 da
(i)
and
12 des.
dx
(ii)
(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
d²
dt2
d²x
= -
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