Solve the Laplace transforms dot x-2ddot x+x=e^t t text{ given } t=0 text{ and } x=0 text{ and } x=1

pedzenekO 2020-11-01 Answered
Solve the Laplace transforms
x˙2x¨+x=ett
 given t=0 and x=0 and x=1
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Expert Answer

Ayesha Gomez
Answered 2020-11-02 Author has 104 answers
Given :
The given Laplace Transform
2x¨+x˙+x=ett
with the given conditions
x(0)=1,x˙(0)=0
Step 2
Solution :
The given Laplace Transform
2x¨+x˙+x=ett(1)
with the given conditions
x(0)=1,x˙(0)=0
Taking Laplace transform of both sides in equation (1) , we have
2[s2x¯sx(0)x(0)]+[sx¯x(0)]+x¯=1(s1)2
2[s2x¯s]+[sx¯1]+x¯=1(s1)2
[2s2+s+1]x¯+[2s1]=1(s1)2
[2s2+2ss+1]x¯=1(s1)2[2s1]
[2s(s1)(s1)]x¯=1(s1)2[2s1]
[(2s1)(s1)]x¯=1(s1)2[2s1]
x¯=1(s1)2[(2s1)(s1)]+[2s1][(2s+1)(s1)]
x¯=1(s1)3(2s1)+[2s1][(2s+1)(s1)]
Solving By Partial fractions ,
[2s1][(2s+1)(s1)]=A(s1)+B(2s+1)
[2s1][(2s+1)(s1)]=A(2s+1)+B(s1)(s1)(2s+1)
2A+B=2,AB=1
3A=1A=13
13B=1B=43
[2s1][(2s+1)(s1)]=13(s1)+43(2s+1)
Now , another term ,
1(s1)3(2s+1)=A(2s+1)+B(s1)+C(s1)2+D(s1)3

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