a) Find the Laplace transformation of the following function by using definition of Laplace transformation: f(t)=cos⁡(pt) where t≥0

Name: a) Find the Laplace transformation of the following function by using definition of Laplace transformation: f(t)=cos⁡(pt) where t≥0
Uploaded: 2022-02-23T17:22:57+00:00
Description: a) Find the Laplace transformation of the following function by using definition of Laplace transformation: f(t)=cos⁡(pt) where t≥0

Question

a) Find the Laplace transformation of the following function by using definition of Laplace transformation:  f(t)=cos⁡(pt) where t≥0

Nathanael Webber · Accepted Answer

Step 1  Given f(t)=cos⁡(pt)  Using the definition of Laplace Transform we have,  L[f(t)]=F(s)=∫0∞e−stf(t)dt  ⇒L[cos⁡pt]=∫0∞e−stcos⁡ptdt  Now we will use Integration by parts to solve it   Let I=∫0∞e−stcos⁡ptdt  I=cos⁡pt∫e−stdt−∫(−psin⁡pt)e−st−sdt  I=cos⁡pt[e−st−s]−ps∫sin⁡pte−stdt  I=−1scos⁡pte−st−ps[sin⁡pt(e−st−s)−∫pcos⁡pt(e−st−s)]  I=[−1scos⁡pte−st−p(s2)sin⁡pte−st−(p2)(s2)I]0∞  ⇒I[1+(p2)(s2)]=−1s[cos⁡pte−st+pssin⁡pte−st]0∞  ⇒I[(s2+p2)s2]=−1s[(0+0)−(1+0)]  ⇒I[(s2+p2)s2]=1s  ⇒I=s(s2+p2)  Step 2  Hence,  I=∫0∞e−stcos⁡ptdt=L[cos⁡pt]=s(s2+p2)  Rightarrow By, definition of Laplace Transform of (f(t)=cos⁡pt) is s(s2+p2)

a) Find the Laplace transformation of the following function by using definition of Laplace transformation:...

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