Reference no: EM131177252
Q 1. Derive the Laplace transform of the sketched f(t) below:
Q 2. Find the inverse Laplace transform of the following functions:
(i) s/L2s2 + n2Π2
(ii) 1/(s + √2)(s - √3)
Q 3. Using Λ[f(t)'] = sF(s) - f(0) , show that Λ[sin2t] = 2/ s(s2 + 4)
Q 4. Solve the following initial value problem by the Laplace transform:
y'' + 2 y' - 3y = 6e-2t, y(0) = 2,
y''(0) = -14.
Q 5. (a) If f(t) has the Laplace transform F(s), then show that
Λ{ f (t - a)u(t - a)} = e-as F(s),
where u(t) is a step function.
(b) Find the inverse transform of e-3s /s3.
Q 6. Using the Laplace transform, solve the following problem.
y'' + 2 y'' - 3y = 8e-t + δ (t - 1/2), y(0) = 3, y'(0) = -5.
Q 7. By differentiation, find the inverse Laplace transform of:
s2 - Π2/(s2 + Π2)2
Q 8. By differentiation, find the Laplace transform of:
t2 coshΠt
Q.9 Using Laplace transforms solve the following integral equation:
y(t) = sin 2t + 0∫t y(t') sin 2(t - t')dt'
Q. 10: Evaluate the following integral:
C∫2z3 + z2 + 4/ z4 + 4z2 dz,
where C is the circle |z - 2| = 4 clockwise.
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