Complex Analysis: Theory and Applications · Problem 11.6.1

Fourier Series and the Laplace Transform — Howell & Matthews

Mathematics — Complex Analysis · laplace-transforms-derivatives-integrals-odes

Problem

Exercises for Section 11.6: 1. Derive

\mathcal{L}(\sin t)

from

\mathcal{L}(\cos t)

. 2. Derive

\mathcal{L}(\cosh t)

from

\mathcal{L}(\sinh t)

. 3. Find

\mathcal{L}(\sin^2 t)

. 4. Show that

\mathcal{L}(te^t) = \frac{1}{(s-1)^2}

. Hint: Let

f(t) = te^t

and

f'(t) = te^t + e^t

. 5. Find

\mathcal{L}^{-1}\!\left(\frac{1}{s(s-4)}\right)

. 6. Find

\mathcal{L}^{-1}\!\left(\frac{1}{s(s^2+4)}\right)

. 7. Show that

\mathcal{L}^{-1}\!\left(\frac{1}{s^2(s+1)}\right) = t - 1 + e^t

. 8. Show that

\mathcal{L}^{-1}\!\left(\frac{1}{s^2(s^2+1)}\right) = t - \sin t

. Solve the initial value problem in the following exercises. 9.

y''(t) + 9y(t) = 0

, with

y(0) = 2

and

y'(0) = 9

. 10.

y''(t) + y(t) = 1

, with

y(0) = 0

and

y'(0) = 2

. 11.

y''(t) + 4y(t) = -8

, with

y(0) = 0

and

y'(0) = 2

. 12.

y'(t) + y(t) = 1

, with

y(0) = 2

. 13.

y'(t) - y(t) = -2

, with

y(0) = 3

. 14.

y''(t) - 4y(t) = 0

, with

y(0) = 1

and

y'(0) = 2

. 15.

y''(t) - y(t) = 1

, with

y(0) = 0

and

y'(0) = 2

. 16.

y'(t) + 2y(t) = 3e^t

, with

y(0) = 2

. 17.

y''(t) + y'(t) - 2y(t) = 0

, with

y(0) = 2

and

y'(0) = -1

. 18.

y''(t) - y'(t) - 2y(t) = 0

, with

y(0) = 2

and

y'(0) = 1