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1.2.16 In Example 1.2.19, we came up with a formula for the half derivatives D/2 (sin(x)) and D'/²(cos(x)). Gener- alize this by coming up with a formula for D" (sin(x)) and D' (cos(x)) where r E R is arbitrary.

1.2.16 In Example 1.2.19, we came up with a formula for the half derivatives D/2 (sin(x)) and D'/²(cos(x)). Gener- alize this by coming up with a formula for D" (sin(x)) and D' (cos(x)) where r E R is arbitrary.

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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1.2.16 In Example 1.2.19, we came up with a formula for the half derivatives D/2(sin(x)) and D'/2(cos(x)). Gener- alize this by coming up with a formula for D" (sin(x)) and D" (cos(x)) where r E R is arbitrary. d.comsbnogober ivotivo mon
Transcribed Image Text:1.2.16 In Example 1.2.19, we came up with a formula for the half derivatives D/2(sin(x)) and D'/2(cos(x)). Gener- alize this by coming up with a formula for D" (sin(x)) and D" (cos(x)) where r E R is arbitrary. d.comsbnogober ivotivo mon
Expert Solution
Step 1

We have to derive the general form of Drsin(x) and Drcos(x)

Since we know that

Dαecx=cαecx     (1)

Now let c=i in (1), then

then we have

sin(x)=eix-e-ix2i

then using (1)

Dαsin(x)=Dαeix-Dαe-ix2i⇒Dαsin(x)=(i)αeix-(-i)αe-ix2i                   (2)

 

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