Question

Evaluate the given Laplace transform without evaluating the integral. L{t2tet}\mathscr{L}\left\{t^{2} * t e^{t}\right\}

Solutions

Verified

Step 1

1 of 2

Laplace transform of given function is

L{t2tet}\mathcal{L}\left\{t^{2}\star te^{t} \right\}

According to convolution theorem 7.97.9

L{fg}=L{f}L{g}=F(s)G(s)\mathcal{L}\left\{ f \star g\right\} =\mathcal{L}\left\{ f\right\}\mathcal{L}\left\{ g\right\}=F(s)G(s)

Where ff and gg be piecewise continuous on [0,)[0,\infty) and of exponential order. Thus

L{t2tet}=L{t2}L{tet}\mathcal{L}\left\{t^{2} \star te^{t} \right\}=\mathcal{L}\left\{t^{2} \right\}\cdot \mathcal{L}\left\{te^{t} \right\}

Since, we know L{tn}=n!sn+1\mathcal{L}\left\{ t^{n}\right\}=\dfrac{n!}{s^{n+1}} and L{eatf(t)}=F(s)ssa\mathcal{L}\left\{ e^{at}f(t)\right\}=F(s)\bigg|_{s \rightarrow s-a}

=2!s31s2ss1=\dfrac{2!}{s^{3}}\cdot \dfrac{1}{s^{2}}\bigg|_{s \rightarrow s-1}

=2!s31(s1)2=\dfrac{2!}{s^{3}}\cdot \dfrac{1}{(s-1)^{2}}

=2s3(s1)2=\dfrac{2}{s^{3}(s-1)^{2}}

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Recommended textbook solutions

Fundamentals of Differential Equations 9th Edition by Arthur David Snider, Edward B. Saff, R. Kent Nagle

Fundamentals of Differential Equations

9th EditionArthur David Snider, Edward B. Saff, R. Kent Nagle
2,119 solutions
Elementary Differential Equations and Boundary Value Problems 10th Edition by Richard C. Diprima, William E. Boyce

Elementary Differential Equations and Boundary Value Problems

10th EditionRichard C. Diprima, William E. Boyce
1,941 solutions
Differential Equations with Boundary-Value Problems 7th Edition by Dennis G. Zill, Michael R. Cullen

Differential Equations with Boundary-Value Problems

7th EditionDennis G. Zill, Michael R. Cullen
2,004 solutions

Related questions