The family of functions y=ce-2x+e-x is solution of the equation y+2y=e-x

Find the constant c which defines the solution which also satisfies the initial condition y (-5) = 6. c=

c = 6*e^ (-10) - e^ (-5) (≈ - e⁻⁵ = - 6.74*10⁻³)

Step-by-step explanation:

for the function

y=c*e^ (-2x) + e^ (-x)

as a solution of y'+2y=e^ (-x)

then for y (x=-5) = 6

6 = c*e^ (-2 (-5)) + e^ ( - (-5)) = c*e^10 + e^5

6 = c*e^10 + e^5

c = (6 - e^5) / *e^10 = 6*e^ (-10) - e^ (-5)

c = 6*e^ (-10) - e^ (-5) (≈ - e⁻⁵ = - 6.74*10⁻³)