Ordinary Differential Equations Multiple Choice Questions on “Laplace Transform of Periodic Function”.
1. Find the laplace transform of f(t), where
f(t) = 1 for 0 < t < a
-1 for a < t < 2a
a) (frac{1}{s} coth(frac{as}{2}))
b) (frac{1}{s} sinh(frac{as}{2}))
c) (frac{1}{s} e^{-as})
d) (frac{1}{s} tanh(frac{as}{2}))
Answer: d
Explanation: In the given question
f(t) is a periodic function having a period 2a
The formula for Laplace Transform is given by:
(L(f(t))=frac{1}{1-e^{-2as}} int_{0}^{2a}e^{-st} f(t)dt)
(L(f(t))=frac{1}{1-e^{-2as}} int_{0}^{a}e^{-st} (1)dt + frac{1}{1-e^{-2as}} int_{a}^{2a}e^{-st}(-1)dt)
=(begin{bmatrix}frac{1}{1-e^(-2as)}×frac{e^{-as}}{-s} – frac{1}{1-e^{-2as}} × frac{-1}{s}end{bmatrix} – begin{bmatrix}frac{1}{1-e^{-2as}} × frac{e^{-2as}}{-s} – frac{1}{1-e^{-2as}} × frac{e^{-as}}{-s}end{bmatrix})
= (frac{1}{1-e^{-2as}}×frac{1}{s}×(1-e^{-as})^2)
= (frac{1}{s}(frac{1+e^{-as}}{1-e^{-as}}))
Dividing both numerator and denominator by (e^{frac{-as}{2}})
= (frac{1}{s} tanh(frac{as}{2}))
Thus, the correct answer is (frac{1}{s} tanh(frac{as}{2})).
2. Find the laplace transform of f(t), where f(t) = |sin(pt)| and t>0.
a) (frac{p}{s^2+p^2}×cosh(frac{spi}{2p}))
b) (frac{p}{s^2+p^2}×sinh(frac{spi}{2p}))
c) (frac{p}{s^2+p^2}×coth(frac{spi}{2p}))
d) (frac{p}{s^2+p^2}×tanh(frac{spi}{2p}))
Answer: c
Explanation: From this question, we know –
Period of sin(t)=2π
Period of sin(pt)=(frac{2pi}{p})
Period of |sin(pt)|=(frac{pi}{p})
(L(f(t))=frac{1}{1-e^{frac{-pi}{ps}}} int_{0}^{frac{pi}{p}}e^{-st} f(t)dt)
Since |sin(pt)| is positive in all quadrants
(L(f(t))=frac{1}{1-e^{frac{-pi}{ps}}} int_{0}^{frac{pi}{p}}e^{-st} sin(pt)dt)
=(frac{1}{1-e^{frac{-pi}{ps}}}begin{bmatrix}frac{e^{frac{-sπ}{p}}}{s^2+p^2}×pend{bmatrix}-begin{bmatrix}frac{1}{s^2+p^2}×(-p)end{bmatrix})
=(frac{1}{1-e^{frac{-pi}{ps}}}×frac{p}{s^2+p^2}×(1+e^{frac{-π}{ps}}))
=(frac{p}{s^2+p^2}×coth(frac{spi}{2p})), (Multiplying and dividing by (e^{frac{-sπ}{2p}}))
Thus, the answer is (frac{p}{s^2+p^2}×coth(frac{spi}{2p})).
Global Education & Learning Series – Ordinary Differential Equations.
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