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}= =∫∞0e−st cosat dt=[e−st(−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 | 1s−a |
sinat | as2+a2 |
cosat | ss2+a2 |
Hyperbolic Function
sinhat=eat−e−at2
coshat=eat+eat2
Example: Find
L{sinhat}
Solution: By the Definition of Laplace Transformation,
L{sinhat}=∫∞0e−stsinhat dt=∫∞0e−st(eat−e−at2) dt=12∫∞0e−st.eat dt−12∫∞0e−st.e−at dt=12L{eat}−12L{e−at}=1s−a−121s+a=122as2−a2=as2−a2
Home Work:
L{coshat}
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