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Lecture 3 MATH 223 (Lt Col Tahmina Sultana)

 

 MATH 223

Lecture 03 

Laplace Transformation





      
    Example: Find  L{cosat}  
Solution:  We know that,  eα.tcosβt dt=eαt(α cosβt+βsinβt)α2+β2  



  L{cosat}=  =0est cosat dt=[est(scosat+asinat)s2+a2]0=[Do It Yourself 🙄]=   =ss2+a2 


    ♦ Some Important Results:  

F(t) L{F(t)}=f(s)
1 1s
t 1s2
t2 2s3
t2 6s4
tn n!sn+1
eat 1sa
sinat as2+a2
cosat ss2+a2







Hyperbolic Function


  sinhat=eateat2 coshat=eat+eat2  

    Example: Find  L{sinhat}   
Solution:  By the Definition of Laplace Transformation,
L{sinhat}=0estsinhat dt=0est(eateat2) dt=120est.eat dt120est.eat dt=12L{eat}12L{eat}=1sa121s+a=122as2a2=as2a2  



    Home Work:  L{coshat}  

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