Question

1) Suppose the solution to the differential equation ( (x-3) y^{prime prime}+3 y=0 ) is written as a power series ( y=sum_{n=0}^{infty} a_{n}(x-1)^{n} ) What is the lower bound of the radius of convergence of this power series? a) ( 0.5 ) b) 1 c) 2 d) 3 e) ( infty ) 2) The Laplace Transform of the function ( f(t)=t-u_{1}(t) t+u_{1}(t)(2-t) ) a) ( left(1-2 mathrm{e}^{-mathrm{s}} ight) / mathrm{s}^{2} ) (b) ( 2 / mathrm{s} ) (c) ( 1 / mathrm{s}^{2}-2 mathrm{e}^{-mathrm{s} / mathrm{s}} ) (d) ( 1 / mathrm{s}^{2}+left(2 / mathrm{s}-1 / mathrm{s}^{2} ight) mathrm{e}^{-mathrm{s}} ) 3) Suppose ( y(t) ) is a solution of the following I.V.P ( y^{prime prime}-2 y^{prime}-y=1, y(0)=-1, y^{prime}(0)=1 ) Then ( mathrm{Y}(mathrm{s})= ) a) ( frac{1}{s^{2}-2 s-1} ) b) ( frac{1}{sleft(s^{2}-2 s-1 ight)} ) c) ( frac{s+1}{s^{2}-2 s-1}+frac{1}{sleft(s^{2}-2 s-1 ight)} ) d) ( frac{-s+3}{s^{2}-2 s-1}+frac{1}{sleft(s^{2}-2 s-1 ight)} ) e) NOTA

          1) Suppose the solution to the differential equation ( (x-3) y^{prime prime}+3 y=0 ) is written as a power series ( y=sum_{n=0}^{infty} a_{n}(x-1)^{n} ) What is the lower bound of the radius of convergence of this power series?
a) ( 0.5 )
b) 1
c) 2
d) 3
e) ( infty )
2) The Laplace Transform of the function ( f(t)=t-u_{1}(t) t+u_{1}(t)(2-t) )
a) ( left(1-2 mathrm{e}^{-mathrm{s}}
ight) / mathrm{s}^{2} )
(b) ( 2 / mathrm{s} )
(c) ( 1 / mathrm{s}^{2}-2 mathrm{e}^{-mathrm{s} / mathrm{s}} )
(d) ( 1 / mathrm{s}^{2}+left(2 / mathrm{s}-1 / mathrm{s}^{2}
ight) mathrm{e}^{-mathrm{s}} )
3) Suppose ( y(t) ) is a solution of the following I.V.P
( y^{prime prime}-2 y^{prime}-y=1, y(0)=-1, y^{prime}(0)=1 )
Then ( mathrm{Y}(mathrm{s})= )
a) ( frac{1}{s^{2}-2 s-1} )
b) ( frac{1}{sleft(s^{2}-2 s-1
ight)} )
c) ( frac{s+1}{s^{2}-2 s-1}+frac{1}{sleft(s^{2}-2 s-1
ight)} )
d) ( frac{-s+3}{s^{2}-2 s-1}+frac{1}{sleft(s^{2}-2 s-1
ight)} )
e) NOTA

$1) Suppose the solution to the differential equation ( (x-3) y^prime prime+3 y=0 ) is written as a power series ( y=sumn=0^infty an(x-1)^n ) What is the lower bound of the radius of convergence of this power series? a) ( 0.5 ) b) 1 c) 2 d) 3 e) ( infty ) 2) The Laplace Transform of the function ( f(t)=t-u1(t) t+u1(t)(2-t) ) a) ( left(1-2 mathrme^-mathrms ight) / mathrms^2 ) (b) ( 2 / mathrms ) (c) ( 1 / mathrms^2-2 mathrme^-mathrms / mathrms ) (d) ( 1 / mathrms^2+left(2 / mathrms-1 / mathrms^2 ight) mathrme^-mathrms ) 3) Suppose ( y(t) ) is a solution of the following I.V.P ( y^prime prime-2 y^prime-y=1, y(0)=-1, y^prime(0)=1 ) Then ( mathrmY(mathrms)= ) a) ( frac1s^2-2 s-1 ) b) ( frac1sleft(s^2-2 s-1 ight) ) c) ( fracs+1s^2-2 s-1+frac1sleft(s^2-2 s-1 ight) ) d) ( frac-s+3s^2-2 s-1+frac1sleft(s^2-2 s-1 ight) ) e) NOTA$

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