Use the method of Laplace transformation to solve initial value problem. \frac{dx}{dt}=x-2y , x(0)=-1, y(0)=2 , \frac{dy}{dt}=5x-y

Falak Kinney 2021-09-14 Answered
Use the method of Laplace transformation to solve initial value problem.
dxdt=x2y,x(0)=1,y(0)=2
dydt=5xy
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Expert Answer

unessodopunsep
Answered 2021-09-15 Author has 105 answers

We know in Laplace transformation
L{yn(t)}=sny¯(s)sn1y(0)sn2y(0)yn1(0)
L{tn}=n!sn+1,L{cosat}=ss2+a2,L{sin(at)}=as2+a2
Now Given IVP is
dxdt=x2y(1)x(0)=1,y(0)=2
dydt=5xy(2)
x(t)x+2y=0
y(t)5x+y=0
Taking laplace transformation on both the equation and both the side
From (1)
L{x(t)}L{x(t)}+2L{y(t)}=0
sx(s)x(0)x(s)+2y(s)=0
(s1)x(s)+2y(s)=1(3)
From (2)
L{y(t)}5L{x(t)}+L{y(t)}=0
sy(s)y(0)5x(s)+y(s)=0
(s+1)y(s)5x(s)=2(4)
5x(s1)x(s)+2y(s)=1
(s1)×(s+1)y(s)5x(s)=2
((s21)+10)y(s)=2(s1)5
y(s)=2(s1)5s2+9=2ss2+9733s2+9
Taking inverse laplace transformation both the side . We get

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