Question
Asked Oct 4, 2019
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What is the transient solution of the following vibrating system:

y'' + 2y' + 2y = cos 2t, y(0) = 0, y'(0) = 3

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Expert Answer

Step 1

Find the complementary solution of the given initial value problem as follows.

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Step 2

Find the particular solution of the given initial value problem as follows.

The particular solution has the form y = Acos 2f+ B sin 2t
y'-24 sin 2 +2B cos 2f
y-4Acos 2t -4Bsin 2/
Substitute in the IVP as follows,
(-4Acos 21 -4Bsin 2r) + 2 (-2.4sin2t +2B cos 2t) +2( cos 2t + Bsin 2r) =cos 21
cos 2r(-4A4B +2.A)+sin 21(-4B - 4A +2 B) = cos 2t
cos 2f (-2A
4B)+ sin21(-2B-4A) =cos 2t
-2.A 4B=1 and -2B-4A = 0
B -2A
-2A 8.A= 1
-104 1
1
and B
10
1
A =
1
cos 2t
10
1
sin 2 t.
The particular solution is y,
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The particular solution has the form y = Acos 2f+ B sin 2t y'-24 sin 2 +2B cos 2f y-4Acos 2t -4Bsin 2/ Substitute in the IVP as follows, (-4Acos 21 -4Bsin 2r) + 2 (-2.4sin2t +2B cos 2t) +2( cos 2t + Bsin 2r) =cos 21 cos 2r(-4A4B +2.A)+sin 21(-4B - 4A +2 B) = cos 2t cos 2f (-2A 4B)+ sin21(-2B-4A) =cos 2t -2.A 4B=1 and -2B-4A = 0 B -2A -2A 8.A= 1 -104 1 1 and B 10 1 A = 1 cos 2t 10 1 sin 2 t. The particular solution is y,

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Step 3

Find the general solut...

y(t)
1
1
C,e cost+Ce sint-cos2t +sin2r
10
5
Apply initial conditions as follows
y(0)=0
1
+(0)-0
10 5
1
G+C(0)
C
10
y'(0)-3
y(t)sin 2cos 2f-e(C, cost+C, sint)+e(-C,sint+C, cost)
2
- (C)+(C,)3
5
2
1
C 3-+
5 10
30 4 1
27
10
10
1
27
1
cos2t+-sin 2t
10
1
y0)
e' sint
cost +
10
10
5
27
sint
COst+
10
1
1
1
cos 2t sin 2
10
= e
10
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Image Transcriptionclose

y(t) 1 1 C,e cost+Ce sint-cos2t +sin2r 10 5 Apply initial conditions as follows y(0)=0 1 +(0)-0 10 5 1 G+C(0) C 10 y'(0)-3 y(t)sin 2cos 2f-e(C, cost+C, sint)+e(-C,sint+C, cost) 2 - (C)+(C,)3 5 2 1 C 3-+ 5 10 30 4 1 27 10 10 1 27 1 cos2t+-sin 2t 10 1 y0) e' sint cost + 10 10 5 27 sint COst+ 10 1 1 1 cos 2t sin 2 10 = e 10

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