. Use the variation of parameters formula together with the exponential from part 3) to find the solution of the IVP in systems form.
. Use the variation of parameters formula together with the exponential from part 3) to find the solution of the IVP in systems form.
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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Do only 4th part
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![Sol.:
Given
ytay't by =e
Substihbing y'= z and yl"= z' we Jet
ス-z-6y=e34
34
2= 2+6y+e3t
Uing the ea-n 7=Z+6y4 e3t. 4 y'=z
on makin orm
"We Con wsile them
let A-L!)
9to ehoyarocleusHe «lg=n i l A-dtl=0 s
ーo→ ニAャ A オA-6A=0
D A2 3, -2
J- も
Eigen veedrs: for A=3
1 A-3I )V, co
-35 =)
A
me ned ati Jind a veeday ų
| -3
S.T
O 0:
ッー;]
au » fim dely mony S2 satirgyehing -2we6y=o
chosen 33·
Those
But we hane
neeod to Find
'a Vredr V2
Po ー2-9 AAI=
•me
S.T (Ar25)% =0
Thde ase
But
we chos 'ų=
Thas, The diagmad makiy D
the esameno of' tho parneipad dingmal and, The Corsinspunding
en
the makm P.
ase endekd
D=
2
to
3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76e48852-165f-4cac-ab1d-f7eea9b92384%2Fe9c3e67c-cb5a-4b64-bce4-4e111333ff28%2F4lq6mb5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Sol.:
Given
ytay't by =e
Substihbing y'= z and yl"= z' we Jet
ス-z-6y=e34
34
2= 2+6y+e3t
Uing the ea-n 7=Z+6y4 e3t. 4 y'=z
on makin orm
"We Con wsile them
let A-L!)
9to ehoyarocleusHe «lg=n i l A-dtl=0 s
ーo→ ニAャ A オA-6A=0
D A2 3, -2
J- も
Eigen veedrs: for A=3
1 A-3I )V, co
-35 =)
A
me ned ati Jind a veeday ų
| -3
S.T
O 0:
ッー;]
au » fim dely mony S2 satirgyehing -2we6y=o
chosen 33·
Those
But we hane
neeod to Find
'a Vredr V2
Po ー2-9 AAI=
•me
S.T (Ar25)% =0
Thde ase
But
we chos 'ų=
Thas, The diagmad makiy D
the esameno of' tho parneipad dingmal and, The Corsinspunding
en
the makm P.
ase endekd
D=
2
to
3
Transcribed Image Text:Problem 3: Consider the IVP
y" + ay' + by = e³*, y(0) = 1, y'(0) = 1.
1. Write the second-order IVP as an IVP of a first-order system of equations by setting
y' = z. Show that the eigenvalue of the resulting matrix are the same as the roots of
the characteristic polynomial that is associated to the second-order IVP.
2. For the remainder of this question, set a = -1 and b = -6.
3. Find the exponential of the matrix of the system by using the diagonalization approach.
4. Use the variation of parameters formula together with the exponential from part 3) to
find the solution of the IVP in systems form.
5. Solve the second-order IVP and compare.
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