Page 130 - Advanced engineering mathematics

110 CHAPTER 3 The Laplace Transform

Put these into the circuit equations to get
t

40i 1 + 120 (i 1 (τ) − i 2 (τ))dτ
0
+ 120(q 1 (0) − q 2 (0)) = 10,
t
60i 2 + 120 i 2 (τ)dτ + 120q 2 (0)
0
t

= 120 (i 1 (τ) − i 2 (τ))dτ + 120(q 1 (0) − q 2 (0)).
0
Setting q 1 (0) = q 2 (0) = 0 in this system, we obtain
t
40i 1 + 120 (i 1 (τ) − i 2 (τ))dτ = 10
0
t t
60i 2 + 120 i 2 (τ)dτ = 120 (i 1 (τ) − i 2 (τ))dτ.
0 0
Apply the transform to obtain
120 120 10
40I 1 + I 1 − I 2 =
s s s
120 120 120
60I 2 + I 2 = I 1 − I 2 .
s s s
These can be written
1
(s + 3)I 1 − 3I 2 =
4
2I 1 − (s + 4)I 2 = 0.
Solve these to obtain
s + 4 3 1 1 1
I 1 (s) = = +
4(s + 1)(s + 6) 20 s + 1 10 s + 6
and
1 1 1 1 1
I 2 (s) = = − .
2(s + 1)(s + 6) 10 s + 1 10 s + 6
Then
3 1
−t
i 1 (t) = e + e −6t
20 10
and
1 1
−t
i 2 (t) = e − e −6t .
10 10
SECTION 3.6 PROBLEMS

In each of Problems 1 through 11, use the Laplace trans- 5. 3x − y = 2t, x + y − y = 0; x(0) = y(0) = 0

form to solve the initial value problem. 6. x + 4y − y = 0, x + 2y = e ; x(0) = y(0) = 0
−t

2
1. x − 2y = 1, x + y − x = 0; x(0) = y(0) = 0 7. x + 2x − y = 0, x + y + x = t ; x(0) = y(0) = 0

2. 2x − 3y + y = 0, x + y = t; x(0) = y(0) = 0 8. x + 4x − y = 0, x + y = t; x(0) = y(0) = 0

3. x + 2y − y = 1,2x + y = 0; x(0) = y(0) = 0 9. x + y + x − y = 0, x + 2y + x = 1; x(0) = y(0) = 0

4. x + y − x = cos(t), x + 2y = 0; x(0) = y(0) = 0 10. x + 2y − x = 0,4x + 3y + y =−6; x(0) = y(0) = 0

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October 14, 2010 14:14 THM/NEIL Page-110 27410_03_ch03_p77-120