Question

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=

Answer 1

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

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)