Solve the differential equation by using Laplace Transform
y''+4y'=15et y(0)=5,y'(0)=-2
Let L { y(t) } = Υ (s)
L{ y'' + 4 y' } = L{15 et } ⇒ L { y'' } +L {4 y' } = 15 L{et} ⇒ L { y'' } + 4L { y' } = 15 L{et} ⇒
s2 Y- s⋅y(0) - y'(0) + 4[ sY⇒ -y(0) ] = 15/(s-1)⇒
s2Y- 5s-2+4sY-20= 15/(s-1)⇒
s2Y + 4sY = 5s +22+ (15/(s-1))⇒
s(s+4)Y= (5s2 +17s - 7) /(s-1) ⇒
Y= (5s2 +17s - 7) /(s(s-1)(s+4))and after the partial fraction decomposition
Y = 7/(4s) + 3/(s-1) + 1/(4(s+4))
Then y(t)= L-1 ( 7/(4s) + 3/(s-1) + 1/(4(s+4)) )
y(t) = 7/4 +3et +e-4t/4