Using Laplace transforms calculate the solution of the ordinary differential equation (d^2 y)(dt^2)-2 (dy)(dt)+y=e^(-t), subject to the initial conditions y(0)=0 and (dy)(dt)(0)=1. Select the correct answer from the following list. a) y(t)=(3te^t)/(2)+(e^(-t))/(4)-(e^t)/(4) b) y(t)=te^t +(t^2 e^t)/(2) c) y(t)=(te^(-t))/(2)-(e^(-t))/(4)+(e^t)/(4) d) y(t)=te^(-t)+(t^2 e^(-t))/(2) e) y(t)=(te^(-t))/(2)-(e^(-t))/(2)+(e^t)/(2)

iarc6io 2022-07-17 Answered
Using Laplace transforms calculate the solution of the ordinary differential equation d 2 y d t 2 2 d y d t + y = e t , subject to the initial conditions y(0)=0 and d y d t ( 0 ) = 1. Select the correct answer from the following list.
a ) y ( t ) = 3 t e t 2 + e t 4 e t 4 b ) y ( t ) = t e t + t 2 e t 2 c ) y ( t ) = t e t 2 e t 4 + e t 4 d ) y ( t ) = t e t + t 2 e t 2 e ) y ( t ) = t e t 2 e t 2 + e t 2
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Answers (1)

sweetwisdomgw
Answered 2022-07-18 Author has 20 answers
d 2 y d t 2 2 d y d t + y = e t       y ( 0 ) = 0 ,       y ( 0 ) = 1
Auxillary equation and given by
m 2 = 2 m + 1 = 0 ( m 1 ) 2 = 0 m = 1 , 1
complementary solution is
y 2 ( x ) = ( c 1 + c 2 t ) e t where c 1 and c 2 are some constant
Particular integral
1 D 2 2 D + 1 e t = 1 ( 1 ) 2 2 ( 1 ) + 1 e t = 1 4 e t
complete solution (y)
y ( t ) = ( c 1 + c 2 t ) e t + 1 4 e t + 1 4 e t
using integral conclusion
y ( 0 ) = c 1 + 1 4 c 1 = 1 4 y ( t ) = ( c 1 + c 2 t ) e t + e t × c 2 1 4 e t y ( 0 ) = c 1 e 0 + c 2 1 4 e 0 c 1 + c 2 1 4 y ( t ) = 1 4 e t + 3 2 t e t + 1 4 e t
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