This is a homogeneous linear differential equation with constant coefficients. The general solution to this kind of equation has the structure
#y(x) = e^(lambda x)#. Substituting
#(lambda^2+6lambda-16)e^(lambda x) = 0#. Here #e^(lambda x) ne 0# so the equation is satisfied for #lambda#'s satisfying
#lambda^2+6lambda-16=0# or #lambda_1=-8, lambda_2=2#
so the solution is
#y = c_1 e^(-8x)+c_2e^(2x)# The constants #c_1,c_2# are choosed according to the initial conditions. So
#y(0)=c_1+c_2=3# and
#y'(0)=-8c_1+2c_2=-4# and finally, solving for #c_1,c_2#
#c_1=-5/3,c_2=14/3# Finally
#y=-5/3e^(-8x)+14/3e^(2x)#
