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(3)= 3.

Answer 1

`c = 1190.2`

Explanation:

We have the following solution:

`y(x) = ce^(-2x) + e^(-x)`

We want to find the value of x that satisfies the following condition:

y(3) = 3.

This means that when x = 3, y = 3. So

`y(x) = ce^(-2x) + e^(-x)`

`y(3) = ce^(-2*3) + e^(-3)`

`3 = ce^(-6) + e^(-3)`

`3 = ce^(-6) + 0.0498`

`ce^(-6) = 2.9502`

`c = (2.9502)/(e^(-6))`

`c = 1190.2`