Identify the characteristic equation, solve for the characteristic roots, and

Gregory Emery 2021-12-27 Answered
Identify the characteristic equation, solve for the characteristic roots, and solve the 2nd order differential equations.
(4D44D323D2+12D+36)y=0
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Jack Maxson
Answered 2021-12-28 Author has 25 answers
To find: The characteristic equation and the solution.
Solution:
Let y=Aemx(A0) be the solution of the given differential equation is
(4D44D323D2+12D+36)y=0
Then the characteristic equation is
4m44m323m2+12m+36=0
(x24x+4)(4x2+12x+9)=0
(x2)2(2x+3)2=0
x=2,2,x=32,32
Then the solution is y(x)=e2x(4+c2x)+e32x(c3+c3x)
Where c1,c2,c3,c4 are arbitrary constant.
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Natalie Yamamoto
Answered 2021-12-29 Author has 22 answers
A.E. is 4m44m323m2+12m+36=0
If m=2,643292+24+36=0
m=2 is a root of inspection. By synthetic division,
i.e.,4m3+4m215m18=0
If m=2
32+163018=0
Again m=2 is a root.
By synthetic division.
4m2+12m+9=0
(2m+3)2=0
m=32,32
The roots of the A.E.
are 2,2,32,32
Thus, y=(C1+C2x)e2x+(C3+C4x)e3x2.
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Vasquez
Answered 2022-01-09 Author has 460 answers

Solution:
(4D44D323D2+12D+36)y=0
the auxilary equation, 4m44m3+23m2+12m+36=0
Use long division we get, m=32 of order 2 and m=2 of order 2
The general solution,
y(x)=Ae32x+Bxe32x+Ce2x+Dxe2x
y(x)=(A+Bx)e32x+(C+Dx)e2x

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