a) State Mellin’s inverse Laplace transform formula.
b) State Cauchy’s residue theorem.
iii. Use (a) and...
a) State Mellin’s inverse Laplace transform formula.
b) State Cauchy’s residue theorem.
iii. Use (a) and (b) to prove that the inverse Laplace transform of
F(s)=1/(s+a) is equal to f(t)= e^(-at),t>0
1) what are Cauchy’s Integral Theorem and Cauchy's integral
formula
2) Explain about the consequences and applications of these
theorems.
3) Explain about different types of singularities and the
difference between Taylor series and Laurent series.
(a) Determine the inverse Laplace transform of F(s) =(2s−1)/s^2
−4s + 6
(b) Solve the initial value problem using the method of Laplace
transform. d^2y/dx^2 −7dy/dx + 10y = 0, y(0) = 0, dy/dx(0) =
−3.
(c) Solve the initial value problem:
1/4(d^2y/dx^2)+dy/dx+4y = 0, y(0) = −1/2,dy/dx(0) = −1.
Use the table of Laplace Transform along with the two properties
to compute the following laplace transforms. (Hint: Trig identities
may be useful for some).
(a) sin(2t)+ cos(4t)
(b) sin(t + π/4)
(c) cosh(t)*sin(t) (recall cosh(t) := e t+e −t 2 )
Use this theorem to find the inverse of the given matrix or show
that no inverse exists. (If an answer does not exist, enter DNE in
any cell.)
1
2
5
1
−1
0
2
1
2
1
−5
0
1
1
2
1
5.State and proof the existence and uniqueness of the Laplace
transform.
6.State and proof the linearity properties of the Laplace
transform.
18.State and proof the first shifting theorem.
Use the Laplace transform to find the solution of the IVP:
a.) 2y' + y = 1, y(0) = 2 (answer should be y(t) = 1 + e-t
/ 2 )
f.) 4y" + y = 0, y(0) = -1, y'(0) = -1 (answer should be y(t) =
-sin(t) - cos(t))
Please show work!