問題1.2 初期値問題

ラプラス変換で初期値問題を解く
L(f’) = sL(f) - y(0)
Y = L(y)

1. y’ + 3y = 10sin t, y(0) = 0
L(sin t) = 1/(s^2 + 1)
を用いる。この方程式の補助方程式は
sY + 3Y = 10 / (s^2 + 1)
(s+3) Y = 10/(s^2 + 1)
Y = 10/(s+3)(s^2 + 1)
Y = 1/(s+3) + 3/(s^2 + 1) - s/(s^2 + 1)
L(y) = L(exp(-3t)) + 3L(sin t) - L(cos t)
L(y) = L(exp(-3t) + 3sin t - cos t)
y = exp(-3t) + 3sin t - cos t

2. y’ - 5y = 1.5 exp(-4t), y(0)=1
L(exp(-4t))=1/(s+4)
を用いる。補助方程式は
sY- 5Y = 1.5/(s+4)
(s-5)Y = 3/2 * 1/(s+4)
Y = 1/6 * {1/(s-5) - 1/(s+4)}
L(y) = 1/6 L(exp(5t)) - 1/6 L(exp(-4t))
y = 1/6 {exp(5t) - exp(-4t)}

3. y’ + 0.2y = 0.01t, y(0) = -0.25
補助方程式は
sY + 0.25 + 0.2Y = 0.01/s^2
(s+0.2)Y = 0.01{1/s^2 - 5^2}
Y = 0.01{1-(5s)^2}/(s+0.2)s^2
Y = 0.05 (1-5s)(1+5s) / (1+5s)s^2
Y = 0.05 (1-5s)/s^2
Y = 0.05/s^2 - 0.25/s
L(y) = 0.05L(t) - 0.25L(1)
L(y) = L(0.05t - 0.25)
y = 0.05t - 0.25

4. y’’ - y’ - 2y = 0, y(0) = 8, y’(0) = 7
{s^2Y - 8s - 7} - {sY- 8} - 2Y = 0
(s^2 - s - 2)Y = 8s - 1
(s-2)(s+1)Y = 8s-1
Y = 3/(s+1) + 5/(s-2)
L(y) = 3L(exp(-t)) + 5L(exp(2t))
L(y) = L(3exp(-t) + 5exp(2t))
y = 3exp(-t) + 5exp(2t)

5. y’’ + ay’ - 2a^2y = 0, y(0)=6, y’(0)=0
{s^2Y - 6s} + a{sY - 6} - 2a^2Y = 0
(s^2 + as - 2a^2)Y = 6s + 6a
(s+2a)(s-a)Y=6(s+a)
Y = 6(s+a) / (s+2a)(s-a)
Y = 2/(s+2a) - 4/(s-a)
L(y) = 2L(exp(-2at)) - 4L(exp(at))
L(y) = L(2exp(-2at) - 4exp(at))
y = 2exp(-2at) - 4exp(at)

6. y’’ + y = 2 cos t, y(0) = 3, y’(0)=4
L(t sin ωt) = 2ωt / (s^2 + ω^2)^2
{s^2Y - 3s - 4} + Y = 2s/(s^2 + 1)
(s^2+1)Y = 2s/(s^2 + 1) + 3s + 4
Y = 2s/(s^2 + 1)^2 +(3s+4)/(s^2 +1)
Y = 2s/(s^2 + 1)^2 + 3 s/(s^2 + 1) + 4 /(s^2 + 1)
L(y) = L(t sin t) + 3L(cos t) + 4L(sin t)
L(y) = L(t sin t + 3cos t + 4 sin t)
y = t sin t + 3 cos t + 4 sin t

7. y’’ - 4y’ + 3y = 6t - 8, y(0) = 0, y’(0) = 0
s^2Y - 4sY + 3Y = 6/s^2 - 8/s
(s^2 - 4s + 3) Y = (6 - 8s)/s^2
(s-1)(s-3)Y = (6-8s)/s^2
Y = (6-8s)/(s-1)(s-3)s^2
Y = 2/s^2 - 2/(s-1)(s-3)
Y = 2/s^2 - 1/(s-3) + 1/(s-1)
L(y) = 2L(t) - L(exp(3t)) + L(exp(t))
L(y) = L(2t - exp(3t) + exp(t))
y = 2t - exp(3t) + exp(t)

8. y’’ + 0.04y = 0.02t^2, y(0) = -25, y’(0) = 0
{s^2Y + 25s} + 0.04Y = 0.02 * 2!/s^3
(s^2 +0.04)Y +25s = 0.04/s^3
(25s^2 + 1)Y + 625s = 1/s^3
(25s^2 + 1)Y = (1- 625s^4)/s^3
Y = (1 + 25s^2)(1 - 25s^2) / s^3(25s^2 + 1)
Y = (1 - 25s^2) / s^3
Y = 1/s^3 - 25/s
L(y) = 1/2 L(t^2) - 25L(1)
L(y) = L(t^2/2 - 25)
y = t^2/2 - 25

9. y’’ + 2y’ - 3y = 6 exp(-2t), y(0) = 2, y’(0) = -14
{s^2Y - 2s +14} + 2{sY - 2} - 3Y = 6/(s+2)
(s^2 + 2s -3)Y = 6/(s+2) +2s +10
(s+3)(s-1)Y = 6/(s+2) +2(s+5)
Y = 6/(s+2)(s-1)(s+3) +2(s+5)/(s-1)(s+3)
Y = 6/(s-1){1/(s+2) - 1/(s+3)} + 2{2/(s+3) - 1/(s-1)}
Y = 6/(s-1)(s+2) - 6/(s-1)(s+3) + 4/(s+3) - 2/(s-1)
Y = 2{1/(s-1) - 1/(s+2)} - 3/2 {1/(s-1) - 1/(s+3)} +4/(s+3) - 2/(s-1)
Y = 2/(s-1) - 2/(s+2) -3/2(s-1) + 3/2(s+3) +4/(s+3) - 2/(s-1)
Y = -3/2(s-1) - 2/(s+2) + 11/2(s+3)
L(y) = -3/2L(exp(t)) -2L(exp(-2t)) + 11/2L(exp(-3t))
L(y) = L{-3exp(t)/2 - 2exp(-2t) +11exp(-3t)/2}
y = -3exp(t)/2 - 2exp(-2t) +11exp(-3t)/2