Find the general solution to the nonhomogeneous equation using the method below: y^2 + 8y + 25 = cos(2t)
Find the general solution to the nonhomogeneous equation using the method below: y^2 + 8y + 25 = cos(2t)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Find the general solution to the nonhomogeneous equation using the method below:
y^2 + 8y + 25 = cos(2t)
![C
[ (D-^5)²¹ F ] ( 1 ) = e^¹)
Find a
gen solution
gen
solution
2²-52 + 6 = 0
(2-3) (2-2)=0
2=3
2=2
or
2+
4 (4)
e
(D-31) (e"+) =
(0-21) (e")=
=
_y₁ (t) = e
4₂ (+) = c²
(D-21) (D-31) y₂ = c²+
(D-31) y = y¹ - 34
6-21) (D-31) y = (D-21) (₁ - 3₁)
e
zd
e
y" - 5y² + by = c²
of
Yp = (D-21) "0 (D-33) (et)
(D-^])² = 0²+5²
-1
e
-75
is
34
f
e
- e
24
y (s) ds
<-34
34
se
L
e
= (y₁ - 3)' - 2 (y1 - 34)
=y" - 34¹-24 1 + 6y
11
=Y - Sy₁ +6y
31
Se
- e
227
e
Sezt
f(s) ds
dt
the
412
e
d+
2
4+
e
ut
e
dt
4+
e
form y(t) = (₁ 4₁ (+) + (₂₂ Y₂ (+) + Yp(+)
dt
Where
DF=f
ut
2
_Yp=
24
gen solution || (+)ace the
K](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe742bcfc-6571-4eb6-8020-0d244ddd1492%2F12aa26df-c1fd-41b8-b1c8-1c489fce25cf%2F3rbz9pl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:C
[ (D-^5)²¹ F ] ( 1 ) = e^¹)
Find a
gen solution
gen
solution
2²-52 + 6 = 0
(2-3) (2-2)=0
2=3
2=2
or
2+
4 (4)
e
(D-31) (e"+) =
(0-21) (e")=
=
_y₁ (t) = e
4₂ (+) = c²
(D-21) (D-31) y₂ = c²+
(D-31) y = y¹ - 34
6-21) (D-31) y = (D-21) (₁ - 3₁)
e
zd
e
y" - 5y² + by = c²
of
Yp = (D-21) "0 (D-33) (et)
(D-^])² = 0²+5²
-1
e
-75
is
34
f
e
- e
24
y (s) ds
<-34
34
se
L
e
= (y₁ - 3)' - 2 (y1 - 34)
=y" - 34¹-24 1 + 6y
11
=Y - Sy₁ +6y
31
Se
- e
227
e
Sezt
f(s) ds
dt
the
412
e
d+
2
4+
e
ut
e
dt
4+
e
form y(t) = (₁ 4₁ (+) + (₂₂ Y₂ (+) + Yp(+)
dt
Where
DF=f
ut
2
_Yp=
24
gen solution || (+)ace the
K
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