Question

Section 8.1 Introduction to the Laplace Transforms

Problem 2.
Use the table of Laplace transforms to find the Laplace transforms of the following functions.

`(a)cosh \: t \: sin \: t`

`(b) {sin}^(2) t`

`(c) {cos}^(2) 2t`

`(d) {cosh}^(2) t`

`(e)t \: sinh \: 2t`

`(f)sin \: t \: cos \: t`

`(g)sin(t + (\pi)/(4) )`

`(h)cos2t - cos3t`

`(i)sin2t + cos4t`
​

Answer 1

I don't know what table you have as reference, but I suspect it includes the following transforms:

`1 \leftrightarrow \frac1s`

`e^(at) \leftrightarrow (1)/(s-a)`

`\cos(at) \leftrightarrow (s)/(s^2+a^2)`

`\sin(at) \leftrightarrow (a)/(s^2+a^2)`

`\cosh(at) \leftrigharrow (s)/(s^2-a^2)`

`\sinh(at) \leftrigharrow (a)/(s^2-a^2)`

It probably also includes some more general properties, like

`t f(t) \leftrightarrow -F'(s)`

`e^(at) f(t) \leftrightarrow F(s-a)`

where F(s) is the Laplace transform of f(t).

Beyond these, you should also know the following identities:

`\cosh(t) = \frac{e^t + e^(-t)}2`

`\cosh^2(t) = \frac{1 + \cosh(2t)}2`

`\cos^2(t) = \frac{1 + \cos(2t)}2`

`\sin^2(t) = \frac{1 - \cos(2t)}2`

`\sin(2t) = 2 \sin(t) \cos(t)`

`\sin(t \pm T) = \sin(t) \cos(T) \pm \cos(t) \sin(T)`

Putting everything together, we have

• (a)

`\cosh(t) \sin(t) = \frac{e^t + e^(-t)}2 * \sin(t) = \frac12 e^t \sin(t) + \frac12 e^(-t) \sin(t)`

and the Laplace transform is

`\frac12 F(s - 1) + \frac12 F(s + 1)`

where F(s) is the transform of sin(t),

`F(s) = (1)/(s^2 + 1)`

Then

`\cosh(t) \sin(t) \leftrightarrow \frac{\frac1{(s-1)^2+1} + \frac1{(s+1)^2+1}}2 = \boxed{(s^2+2)/(s^4+4)}`

• (b)

`\sin^2(t) = \frac12 \left(1 - \cos(2t)\right)`

and the transform is

`F(s) = \frac12 \left(\frac1s - (s)/(s^2+4)\right) = \boxed{(2)/(s^3+4s)}`

• (c)

`\cos^2(2t) = \frac12 \left(1 + \cos(4t)\right)`

with transform

`F(s) = \frac12 \left(\frac1s + (s)/(s^2+16)\right) = \boxed{(s^2+8)/(s^3+16s)}`

• (d)

`\cosh^2(t) = \frac12 \left(1 + \cosh(2t)\right)`

with transform

`F(s) = \frac12 \left(\frac1s + (s)/(s^2-4)\right) = \boxed{-\frac2{s^3-4s}}`

• (e)

`t\sinh(2t) \leftrightarrow -F'(s)`

where F(s) is the Laplace transform of sinh(2t),

`F(s) = (2)/(s^2 - 4) \implies -F'(s) = \boxed{(4s)/((s^2-4)^2)}`

• (f)

`\sin(t) \cos(t) = \frac12 \left(2\sin(t) \cos(t)\right) = \frac12 \sin(2t)`

with transform

`F(s) = \frac12 * (2)/(s^2+4) = \boxed{\frac1{s^2+4}}`

• (g)

`\sin\left(t+\frac\pi4\right) = \sin(t) \cos\left(\frac\pi4\right) + \cos(t) \sin\left(\frac\pi4\right) = \frac1{\sqrt2} \left(\sin(t) + \cos(t)\right)`

with transform

`F(s) = \frac1{\sqrt2} \left(\frac1{s^2+1} + (s)/(s^2+1)\right) = \boxed{(s+1)/(\sqrt2 (s^2+1))}`

The last two are trivial and follow directly from the properties listed above.

• (h)

`\cos(2t) - \cos(3t) \leftrightarrow (s)/(s^2+4) - (s)/(s^2+9) = \boxed{(5s)/(s^4 + 13s^2 + 36)}`

• (i)

`\sin(2t) + \cos(4t) \leftrightarrow \frac2{s^2+4} + (s)/(s^2+16) = \boxed{(s^3+2s^2+4s+32)/(s^4+20s^2+64)}`