the differential equation be solved using the Laplace Transform? A=3 x^{II}(t)-3x^{I}(t)+2x(t)=A e^{2t} x(0)=-3 x^{I}(0)=5

CheemnCatelvew 2020-12-01 Answered
the differential equation be solved using the Laplace Transform? A=3
xII(t)3xI(t)+2x(t)=Ae2t
x(0)=3
xI(0)=5
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Expert Answer

averes8
Answered 2020-12-02 Author has 92 answers
Step 1
Some Laplace Transforms,
1.L(x(t))=s2x(s)sx(0)x(0)
2.L(x(t))=sx(s)x(0)
3.L(x(t))=x(s)
L1(x(s))=x(t)
4.L(eat)=1(sa) then L(e2t)=1(s2)
L1(1(sa))=eat
5.L(teat)=1(sa)2L1(1(sa)2)=teat
Step 2
Given differential equation is
x(t)3x(t)+2x(t)=3e2t(as given A=3) and x(0)=3,x(0)=5
Take Laplace Transform of every term of given differential equation, we get
L(x(t))3L(x(t))+2L(x(t))=3L(e2t)
using step 1, we get
s2x(s)sx(0)x(0)3(sx(s)x(0))+2x(s)=3(1(s2))
as given initial conditions are
x(0)=3,x(0)=5 then 
s2x(s)s(3)53(sx(s)(3))+2x(s)=3(1(s2))
s2x(s)+3s53sx(s)9+2x(s)=3(1(s2))
s2x(s)3sx(s)+2x(s)+3s14=3(s2)
(s23s+2)x(s)+3s14=3(s2)
(s2)(s1)x(s)=3(s2)3s+14
x(s)=3(s2)2(s1)(3s14)(s2)(s1)
Step 3
Partial fractional decomposition
(3s14)(s2)(s1)=11(s1)8(s2)
3(s2)2(s1)=3(s1)3(s2)+3(s2)2
then equation (1) becomes

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