y6y25y0 y0 when xo and ye3pi8 when x pi8SolutionThe charact

y\"+6y\'+25y=0 ; y=0 when x=o and y=e^(-3pi/8) when x= pi/8

The characteristic equation is:

r^2 + 6r + 25 = 0

(r+3)^2 + 4^2 = 0

The roots are:

r = -3+4i , -3-4i

So the general solution is:

y = e^(-3x) (Asin(4x) + Bcos(4x))

y(0) = B = 0

y(/8) = e^(-3/8)(A*sin(/2)) = A.e^(-3/8) = e^(-3/8) -> A = 1

So the particular solution is:

y = e^(-3x).sin(4x)

$y\$