## 17Calculus - Laplace Transforms Involving Derivatives and Integrals

##### 17Calculus

We can apply the Laplace Transform integral to more than just functions. We can also do transformations to equations involving derivatives and integrals. This will allow us to solve differential equations using Laplace Transforms.

The Laplace Transform equations involving a derivative or integral are not hard to derive but they do use techniques that you might not consider. Let's look at three in particular and watch videos on deriving their formulas.

Laplace Transforms of Derivatives

Let's start with the Laplace Transform of $$f'(t)$$.

$$\mathcal{L}\{ f'(t) \} = sF(s) - f(0)$$

In this formula, $$\mathcal{L}\{ f(t) \} = F(s)$$ and $$f(0)$$ is the initial condition. Here is the video that derives this formula.

### blackpenredpen - Laplace Transform of the Derivative [10mins-28secs]

video by blackpenredpen

The second derivative is similar.

$$\mathcal{L}\{ f''(t) \} = s^2F(s) - sf(0) - f'(0)$$

### blackpenredpen - Laplace Transform of the Derivative [10mins-28secs]

video by blackpenredpen

From the second derivative, the nth-derivative can be easily extrapolated.

Laplace Transforms of Integrals

This video clip shows the derivation of $$\mathcal{L}\{ \int_0^t{ f(v)~dv } \}$$ using the integral definition of the Laplace Transform.

### blackpenredpen - Laplace Transform of the Derivative [10mins-28secs]

video by blackpenredpen

Laplace Transforms Table

### Laplace Transforms Table

Laplace Transforms

$$f(t)$$

$$\displaystyle{ F(s) }$$

Basic Functions

$$t^n, ~ n = 1, 2, 3, \ldots$$

$$\displaystyle{ \frac{n!}{s^{n+1}} }$$

$$e^{at}$$

$$\displaystyle{ \frac{1}{s-a} }$$

$$\sin(\alpha t)$$

$$\displaystyle{ \frac{\alpha}{s^2 + \alpha^2} }$$

$$\cos(at)$$

$$\displaystyle{ \frac{s}{s^2 + a^2} }$$

$$\sinh(at)$$

$$\displaystyle{ \frac{a}{s^2 - a^2} }$$

$$\cosh(at)$$

$$\displaystyle{ \frac{s}{s^2 - a^2} }$$

Special Functions

$$\delta(t)$$ unit impulse

$$1$$

$$\delta(t-\tau)$$ shifted unit impulse

$$e^{-\tau s}$$

$$u(t)$$ unit step

$$\displaystyle{ \frac{1}{s} }$$

$$u(t-\tau)$$ shifted unit step

$$\displaystyle{ \frac{1}{s} e^{-\tau s} }$$

Combined Functions

$$e^{at}\sin(\alpha t)$$

$$\displaystyle{ \frac{\alpha}{(s-a)^2 + \alpha^2} }$$

$$f(t)u(t-a)$$

$$\displaystyle{ e^{-sa} \mathcal{L}\{ f(t+a) \} }$$

$$t^n e^{at}, ~ n = 1, 2, 3, \ldots$$

$$\displaystyle{ \frac{n!}{(s-a)^{n+1}} \} }$$

Derivatives and Integrals

$$f'(t)$$

$$sF(s) - f(0)$$

$$f''(t)$$

$$s^2F(s) - sf(0) - f'(0)$$

$$\displaystyle{ f^{(n)}(t) }$$

$$\displaystyle{ s^nF(s) - s^{n-1}f(0) - }$$ $$\displaystyle{ s^{n-2}f'(0) - . . . - f^{(n-1)}(0) }$$

$$\int_0^t{ f(v)~dv }$$

$$\displaystyle{ \frac{F(s)}{s} }$$

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