2 The Laplace transform of a derivative

Here we consider not a causal function $f\left(t\right)$ directly but its derivatives $\frac{df}{dt}$ , $\frac{d^{2}f}{d{t}^2}$ , …(which are also causal.) The Laplace transform of derivatives will be invaluable when we apply the Laplace transform to the solution of constant coefficient ordinary differential equations.

If $\mathcal{L}\left\{f\left(t\right)\right\}$ is $F\left(s\right)$ then we shall seek an expression for $\mathcal{L}\left\{\frac{df}{dt}\right\}$ in terms of the function $F\left(s\right)$ .

Now, by the definition of the Laplace transform

$\mathcal{L}\left\{\frac{df}{dt}\right\}={\int \; <!--nolimits\; -->}_0^{\infty }{\text{e}}^{-st}\frac{df}{dt}dt$

This integral can be simplified using integration by parts:

(As usual, we assume that contributions arising from the upper limit, $t=\infty$ , are zero.) The integral on the right-hand side is precisely the Laplace transform of $f\left(t\right)$ which we naturally replace by $F\left(s\right)$ . Thus

$\mathcal{L}\left\{\frac{df}{dt}\right\}=-f\left(0\right)+sF\left(s\right)$

As an example, we know that if $f\left(t\right)=sintu\left(t\right)$ then

$\mathcal{L}\left\{f\left(t\right)\right\}=\frac{1}{s^2+1}=F\left(s\right)\text{ (Table 1, Rule 5)}$

and so, according to the result just obtained,

a result we know to be true.

We can find the Laplace transform of the second derivative in a similar way to find:

$\mathcal{L}\left\{\frac{d^{2}f}{d{t}^2}\right\}=-f^{\prime }\left(0\right)-sf\left(0\right)+s^{2}F\left(s\right)$

(The reader might wish to derive this result.) Here $f^{\prime }\left(0\right)$ is the derivative of $f\left(t\right)$ evaluated at $t=0$ .

Task!

If $\mathcal{L}\left\{f\left(t\right)\right\}=F\left(s\right)\text{ and}\frac{d^{2}f}{d{t}^2}-\frac{df}{dt}=3t$ with initial conditions

$f\left(0\right)=1,f^{\prime }\left(0\right)=0$ , find the explicit expression for $F\left(s\right)$ .

Begin by finding $\mathcal{L}\left\{\frac{d^{2}f}{d{t}^2}\right\},\mathcal{L}\left\{\frac{df}{dt}\right\}\text{ and}\mathcal{L}\left\{3t\right\}$ :

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Exercises

1. Find the Laplace transforms of
   1. $t^3{\text{e}}^{-2t}u\left(t\right)$
   2. ${\text{e}}^{t}sinh3t.u\left(t\right)$
   3. $sin\left(t-3\right).u\left(t-3\right)$
2. If $F\left(s\right)=\mathcal{L}\left\{f\left(t\right)\right\}$ find expressions for $F\left(s\right)$ if
   1. $\frac{d^{2}y}{d{t}^2}-3\frac{dy}{dt}+4y=sinty\left(0\right)=1,y^{\prime }\left(0\right)=0$
   2. $7\frac{dy}{dt}-6y=3u\left(t\right)y\left(0\right)=0,$
3. Find the inverse Laplace transforms of
   1. $\frac{6}{{\left(s+3\right)}^4}$
   2. $\frac{15}{s^2-2s+10}$
   3. $\frac{3s^2+11s+14}{s^3+2s^2-11s-52}$
   4. $\frac{{\text{e}}^{-3s}}{s^4}$
   5. $\frac{{\text{e}}^{-2s-2}\left(s+1\right)}{s^2+2s+5}$

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