Properties of Laplace Transform

 $Laplace\ transform\ of\ e^{3t}-\frac{1}{2}$  

•  $\frac{1}{s+3}-\frac{1}{2s}$  

•  $\frac{1}{s-3}-\frac{2}{s}$  

•  $\frac{1}{s-3}-\frac{1}{2s}$  

•  $\frac{1}{2\left(s-3\right)}$  

 $L\left[e^{-2t}\sin3t\right]$  

•  $L\left[\sin3t\right],\ s\rightarrow s+2$  

•  $L\left[\sin3t\right],\ s\rightarrow s-2$  

 $f\left(t\right)=te^{-2t}$  

• 
• 
• 
• 

 $L\left[e^{5t}\cos3t\right]$  

• 
• 
• 
• 

Define this property:  $L\left(f\left(t\right)e^{at}\right)=F\left(s-a\right)$  

• Linearity Property

• First Shifting Property

• Convolution Theorem

• Second Shifting Property

 $L\left(e^{2t\ }\sin\ t\right)$  What is  $a$   and f(t)?

•  $a=2,\ f\left(t\right)=e^{2t}$  

•  $a=4,\ f\left(t\right)=\cos\ t$  

•  $a=2,\ f\left(t\right)=\sin\ t$  

Find  $L\left(e^{3t}\left(t^3-3t^2+5t\right)\right)$  

•  $\frac{2}{\left(s-3\right)^4}-\left(\frac{3}{\left(\left(s-3\right)^3\right)}\right)+\frac{5}{\left(s-3\right)^2}$  

•  $\frac{6}{\left(s-3\right)^4}-\left(\frac{6}{\left(s-3\right)^3}\right)+\frac{5}{\left(s-3\right)^2}$  

•  $\frac{2}{\left(s-3\right)^3}-\left(\frac{3}{\left(s-3\right)^{\left(2\right)}}\right)+\frac{5}{s-3}$  

Tags

CCSS.HSA.APR.A.1

CCSS.HSA.SSE.A.2

CCSS.HSA.SSE.B.3

 $L\left(t\ \cos\ 6t\right)$  Which Laplace property is use to solve this?

• Linearity Property

• Derivative of Laplace Transform

• First Shifting Property

• Second Shifting Property

 $L\left(t\ \sin\ 4t\right)$  What is n, f(t) and L[f(t)]?

•  $n=1,\ f\left(t\right)=\sin\ 4t,\ F\left(s\right)=\frac{4}{s^2+16}$  

•  $n=4,\ f\left(t\right)=t,\ F\left(s\right)=\frac{4}{s^2+16}$  

•  $n=1,\ f\left(t\right)=\cos\ 6t,\ F\left(s\right)=\frac{s}{s^2+14}$  

What is the formula for Laplace First Order Derivative?

• $ℒ\left(𝑓′(𝑡)\right)=𝑠𝐹(𝑠)−𝑓(0)$

• $ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑠𝑓\left(0\right)−𝑓′\left(0\right)$

• $ℒ\left(𝑓′(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$