Use the Laplace transform to solve the given initial value problem.y"+2y'+5y=0;y

OlmekinjP 2021-10-14 Answered
Use the Laplace transform to solve the given initial value problem.y"+2y'+5y=0;y(0)=2,y'(0)=−1
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Expert Answer

yagombyeR
Answered 2021-10-15 Author has 92 answers
s2Ysy(0)y(0)+2sY2y(0)+5Y=0 Take the Laplace transform of both sides.
Y(s2+2s+5)2s+14=0 Substitute in the intial conditions and solve for Y.
Y((s+1)2+4)=2s+3
Y=2s+3(s+1)2+4
Y=2(s+11)+3(s+1)2+4 Add 1 to the s on top so that it matches the one on the bottom of the fraction, then split the fraction into two.
Y=2(s+1)+1(s+1)2+4
Y=2(s+1)(s+1)2+4+1(s+1)2+4
y=2etcos2t+12etsin2t Take the inverse Laplace transform
Result:
y=2etcos2t+12etsin2t
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