Show that L\left\{\sin kt\right\}=\frac{k}{s^2+k^2} , L\left\{\cos kt\right\}=\frac{s}{s^2+k^2}

alesterp

alesterp

Answered question

2021-09-13

Show that L{sinkt}=ks2+k2
L{coskt}=ss2+k2
Use L{sinkt}=0estsinktdt
1)L{1}=
2)L{tn}=
3)L{ekt}=
4)L{sinkt}=
5)L{coskt}=
6)L{sinhkt}=
7)L{coshkt}=

Answer & Explanation

Aamina Herring

Aamina Herring

Skilled2021-09-14Added 85 answers

Step 1
We have to lapace transform of sinkt and coskt
Step 2
Show that L{sinkt}=ks2+k2
L{coskt}=ss2+k2
Solution: We know that
L{ f(t)}=0estf(t)dt
Now, L{sinkt}=0estsinktdt
=L{sinkt}=0estsinktdt
=[est((s)2+k2)(ssinktkcoskt)]0
=0e0(s2+k2)(0k)=ks2+k2
Hence L{sinkt}=ks2+k2
Again L{coskt}=0estcosktdt
=[est((s)2+k2)(scoskt+ksinkt)]0
=0(e0s2+k2(s+0))=ss2+k2
Hence , L{coskt}=s(s2+k2)
Note:
(1)eaxsinbxdx=eaxa2+b2(asinbxbcosbx)
(2)eaxcosbxdx=eaxa2+b2(acosbx+bsinbx)
(3)L{1}=13
(4)L{tn}=n!sn+1
(5)

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