If y(x) = e^-3x , find A so that satisfies the following equation

Question

d^2/dx^2 y(x) + 2 (d/dx y(x)) + A y(x) = 0

Answer 1

Since y(x)=e^(-3x) we obtain that y'(x)=-3e^(-3x) and y''(x)=9e^(-3x). Plugging this into the differential equation

y''+2y'+Ay=0 we arrive at 9e^(-3x)-6e^(-3x)+Ae^(-3x)=0 which is the same as (3+A)e^(-3x)=0. But an exponential function can never be zero. Thus, we must have A=-3 which is the answer.