Find y as a function of t if 4y 729y 0 with y0 2 y0 9 y

Find y as a function of t if 4y\'\' - 729y = 0 with y(0) = 2, y\'(0) = 9. y =

Solution

Let y = e^rt be the solution of the given differential equal 4y\'\' - 729y = 0

=> 4r² - 729 = 0

=> r₁ = 27/2 and r₂ = -27/2

Therefore , the solution of the differential equation would be

y = c₁e^(27/2)t + c₂e^(-27/2)t

y\' = (27/2)( c₁e^(27/2)t - c₂e^(-27/2)t )

y(0) = c₁ + c₂ = 2

y\'(0) = (27/2)( c₁ - c₂ ) = 9 => ( c₁ - c₂ ) = 2/3

=> c₁ = 4/3 and c₂ = 2/3

=> y = (4/3)e^(27/2)t + (2/3)e^(-27/2)t