Solve the differential equation by using Laplace Transform

Question

Solve the differential equation by using Laplace Transform

y''+4y'=15e^t y(0)=5,y'(0)=-2

Answer 1

Solve the differential equation by using Laplace Transform

y''+4y'=15e^t y(0)=5,y'(0)=-2

Let L { y(t) } = Υ (s)

L{ y^'' + 4 y^' } = L{15 e^t } ⇒ L { y^'' } +L {4 y^' } = 15 L{e^t} ⇒ L { y^'' } + 4L { y^' } = 15 L{e^t} ⇒

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 +3e^t +e^-4t/4