Using Laplace Transform with initial values y01 y08 solve y

Using Laplace Transform with initial values y(0)=1 , y\'(0)=8 solve y\'\'+4y\'=0

Solution

r² + 4 = 0, r₁ = 2i, r₂ = 2i

y(t) = e^0t [C₁ cos(2t) + C₂ sin(2t)] = C₁ cos(2t) + C₂ sin(2t)

y \'(t) = 2C₁ sin(2t) + 2C₂ cos(2t)

C₁ cos(0) + C₂ sin(0) = 1, C₁ = 1

2C₁ sin(0) + 2C₂ cos(0) = 8, C₂ = 4

y(t) = cos(2t) + 4sin(2t)