(g) Give an example of an IVP in which the formula you found in part (f) helps you find a real solution. Solve the IVP.
(g) Give an example of an IVP in which the formula you found in part (f) helps you find a real solution. Solve the IVP.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I only need part g
Transcribed Image Text:(Euler's Formula) Consider f(t) = e(a+iß)t, where a, ß E R.
(a) Rewrite f(t) as a product of a real-valued function and a complex-valued function using
properties of exponents.
(b) Rewrite the complex-valued function in part (a) using the Maclaurin series e" =
Lk=0 k! ·
Write at least the first six terms of the series.
(c) Next, simplify any factors i", n > 2, that you may have written in part (b).
(d) The series written in (c) is absolutely convergent, so we may re-order the terms. Re-order
the terms in the sum by grouping all the even terms together and then grouping all the
odd terms together.
(e) Write the complex number in the form fi(t) + i f2(t) where fi (t) is an even, real-valued
function and f2(t) is an odd, real-valued function. (Hint: Recall the Maclaurin series for
two other elementary functions)
(f) Rewrite f(t) using the formula you found in (e).
(g) Give an example of an IVP in which the formula you found in part (f) helps you find a
real solution. Solve the IVP.
Transcribed Image Text:(f).
Since the given function f (t) = elatif" can be written as f (t) = fi (t) + if2 (t), where:
fi (t)=e'“ cos (tß)
f2 (1)=e'“ sin (tß)
Therefore,
f (t)=fi (t) + if2 (t)
=e" cos (tB) + ie“ sin (tß)
=et" (cos (tB) + i sin (tß))
Hence elatiß)t =e« (cos (tß) + i sin (tß)) ·
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