Question

1. Find the Laplace transform and the inverse Laplace trans of the following functions (40 Points) a) t ? f_1(t) = 1 + t - 2??t + (e^2t - e^-2t) / 2 b) t ? f_3(t) = t^n e^at c) L{f(t)} = { 5, 0 < t < 2; -3, 2 < t < 6; 0, 6 ? t d) L^-1{1/(??s + 3)} e) t ? f_2(t) = e^at - 2/???t + e^3t sin 4t f) t ? f_4(t) = sin^2(3t) g) L^-1{(s+10)/((s-3)^2 + 4)} h) L^-1{(5s^2+20)/(s(s-1)(s^2+5s+4))} 2. Using Laplace transform definition prove the following (20 Points) a) L{e^at} = 1/(s-a) b) L{y'(t)} = s Y(s) - y(0) c) L{sin t} = a/(s^2+a^2) d) L{c_1 f(t) + c_2 f_2(t)} = c_1 F(s) + c_2 F_2(s) 3. Solve the following IVP using the Laplace transform (30 points) a) y'(t) + y(t) = e^3t, y(t=0) = 0 b) y'' - y = 8t, y(0) = 2 and y'(0) = -2 c) y''(t) + 2y(t) = f(t), y(0) = y'(0) = 0, f(t) = { 1, 0 < t < ??; 0, ?? < t < 2??; sin t, t > 2??

          1. Find the Laplace transform and the inverse Laplace trans of the following functions (40 Points)
a) t ? f_1(t) = 1 + t - 2??t + (e^2t - e^-2t) / 2
b) t ? f_3(t) = t^n e^at
c) L{f(t)} = { 5, 0 < t < 2; -3, 2 < t < 6; 0, 6 ? t
d) L^-1{1/(??s + 3)}
e) t ? f_2(t) = e^at - 2/???t + e^3t sin 4t
f) t ? f_4(t) = sin^2(3t)
g) L^-1{(s+10)/((s-3)^2 + 4)}
h) L^-1{(5s^2+20)/(s(s-1)(s^2+5s+4))}
2. Using Laplace transform definition prove the following (20 Points)
a) L{e^at} = 1/(s-a)
b) L{y'(t)} = s Y(s) - y(0)
c) L{sin t} = a/(s^2+a^2)
d) L{c_1 f(t) + c_2 f_2(t)} = c_1 F(s) + c_2 F_2(s)
3. Solve the following IVP using the Laplace transform (30 points)
a) y'(t) + y(t) = e^3t, y(t=0) = 0
b) y'' - y = 8t, y(0) = 2 and y'(0) = -2
c) y''(t) + 2y(t) = f(t), y(0) = y'(0) = 0, f(t) = { 1, 0 < t < ??; 0, ?? < t < 2??; sin t, t > 2??

1. Find the Laplace transform and the inverse Laplace trans of the following functions (40 Points)
a) t ? f1(t) = 1 + t - 2??t + (e^2t - e^-2t) / 2
b) t ? f3(t) = t^n e^at
c) Lf(t) = 5, 0 < t < 2; -3, 2 < t < 6; 0, 6 ? t
d) L^-11/(??s + 3)
e) t ? f2(t) = e^at - 2/???t + e^3t sin 4t
f) t ? f4(t) = sin^2(3t)
g) L^-1(s+10)/((s-3)^2 + 4)
h) L^-1(5s^2+20)/(s(s-1)(s^2+5s+4))
2. Using Laplace transform definition prove the following (20 Points)
a) Le^at = 1/(s-a)
b) Ly'(t) = s Y(s) - y(0)
c) Lsin t = a/(s^2+a^2)
d) Lc1 f(t) + c2 f2(t) = c1 F(s) + c2 F2(s)
3. Solve the following IVP using the Laplace transform (30 points)
a) y'(t) + y(t) = e^3t, y(t=0) = 0
b) y” - y = 8t, y(0) = 2 and y'(0) = -2
c) y”(t) + 2y(t) = f(t), y(0) = y'(0) = 0, f(t) = 1, 0 < t < ??; 0, ?? < t < 2??; sin t, t > 2??

Added by Keith B.

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