Question

In Problems 19–36 use theorem 7.1.1 to find$L\left\{f\left(t\right)\right\}.$

25.$f\left(t\right)={\left(t+1\right)}^3$

Short answer

The Laplace transform of given function is,

$L\left\{{\left(t+1\right)}^3\right\}=\frac{6}{s^4}+\frac{6}{s^3}+\frac{3}{s^2}+\frac{1}{s},s>0$

Step-by-step solution

Definition

Here we determine the Laplace transform of the given function by comparing it to general form as provided in theorem 7.1.1.

$L\left\{f\left(t^n\right)\right\}=\frac{n!}{s^{n+1}},s>0$

Solution

Consider the function$f\left(t\right)={\left(t+1\right)}^3$

The objective is to find $L\left\{f\left(t\right)\right\}$using the theorem.

From the linearity property of the Laplace transform, we have

$$\begin{gathered} L\left\{f\left(t\right)\right\}={\left(t+1\right)}^3=t^3+3t^2+3t+1 \\ L\left\{f\left(t\right)\right\}=L\left\{t^3\right\}+3L\left\{t^2\right\}+3L\left\{t\right\}+L\left\{1\right\} \end{gathered}$$

From the theorem7.1.1,

$L\left\{f\left(t^n\right)\right\}=\frac{n!}{s^{n+1}},s>0$

$L\left\{f\left(t^n\right)\right\}=\frac{n!}{s^{n+1}},s>0$; Where$n=0,1,2,3,............$

$$\begin{gathered} L\left\{{\left(t+1\right)}^3\right\}=\left\{\frac{3!}{s^{3+1}}\right\}+3\left\{\frac{2!}{s^{2+1}}\right\}+3\left\{\frac{1!}{s^{1+1}}\right\}+L\left\{\frac{0!}{s^{0+1}}\right\} \\ L\left\{{\left(t+1\right)}^3\right\}=\frac{6}{s^4}+\frac{6}{s^3}+\frac{3}{s^2}+\frac{1}{s},s>0 \end{gathered}$$

Therefore the required Laplace transform of function is,

$L\left\{{\left(t+1\right)}^3\right\}=\frac{6}{s^4}+\frac{6}{s^3}+\frac{3}{s^2}+\frac{1}{s},s>0$