(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
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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
= -
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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