What is the general solution of the differential equation? : d^4y/dx^4−4y=0

Leonel Schwartz 2022-09-29 Answered
What is the general solution of the differential equation? : d 4 y d x 4 - 4 y = 0
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Answers (1)

Bleha7s
Answered 2022-09-30 Author has 11 answers
With linear operator D = d d x , this is:
( D 4 - 4 ) y = 0
( D 2 - 2 ) ( D 2 + 2 ) y = 0
( D - 2 ) ( D + 2 ) ( D - i 2 ) ( D + i 2 ) y = 0
Each of these factors has a simple solution, eg:
1. ( D - 2 ) y = 0 y = 2 y y = α e 2 x
2. ( D - i 2 ) y = 0 y = i 2 y y = β e i 2 x
The complete solution is the superposition of individual solutions:
y ( x ) = A e 2 x + B e - 2 x + C e i 2 x + D e - i 2 x
C e i 2 x + D e - i 2 x can also be re-written using the Euler formula , arriving at:
y = A e 2 x + B e - 2 x + C cos ( 2 x ) + D sin ( 2 x )
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