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

Question

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

Answer 1

If y = e^-x, then dy/dx = -e^-x and d²y/dx² = e^-x

So, we have e^-x + e^-x + Ae^-x = 0

e^-x(2 + A) = 0. Since e^-x is never 0, 2 + A = 0. So, A = -2.