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)y(x)=eλx. Substituting
(lambda^2+6lambda-16)e^(lambda x) = 0(λ2+6λ−16)eλx=0. Here e^(lambda x) ne 0eλx≠0 so the equation is satisfied for lambdaλ's satisfying
lambda^2+6lambda-16=0λ2+6λ−16=0 or lambda_1=-8, lambda_2=2λ1=−8,λ2=2
so the solution is
y = c_1 e^(-8x)+c_2e^(2x)y=c1e−8x+c2e2x The constants c_1,c_2c1,c2 are choosed according to the initial conditions. So
y(0)=c_1+c_2=3y(0)=c1+c2=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)
