Find the general solution of the given differential equation. y^{(6)}+y=0

balff1t

balff1t

Answered question

2021-11-14

Find the general solution of the given differential equation.
y(6)+y=0

Answer & Explanation

Charles Clute

Charles Clute

Beginner2021-11-15Added 17 answers

Given:
y(6)+y=0
Determine the characteristic equation by replacing y6 with r6 and y with 1 in the differential equation:
r6+1=0
Subtract 1 from each side:
r6=1
Take the 6th root of each side:
r=16=eikπ6=coskπ6+isinkπ6
The corresponding six roots are then 32±12i,32±12i and ±i
The general solution for two each pair of complex roots is then:
y=c1eatcosbt+c2eatsinbt
with a the real part of the roots and b the imaginary part of the roots.
The general solution then becomes:
y(t)=c1e3t2cos12t+c2e3t2sin12t+c3e3t2cos12t+c4e3t2sin12t+c5cost+c6sint

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