Question

Find a such that the solution of y'' + y' − 2y = 0, y(0) = a, y' (0) = 1 tends to zero as t → +∞.

Answer 1

Explanation:

`y'' + y' − 2y = 0, y(0) = a, y' (0) = 1`

Auxialary equation is

`m^2+m-2=0\\m=-2,1`

General solution is

`y=Ae^(-2x) +Be^x`

`y(0) = A+B =a`

`y'(0) = -2Ae^(2x) +Be^x = -2A+B = 1`

Eliminate B to get

3A =a-1

We know that y tends to 0 when x tends to infinity for any finite A

i.e. a should be a finite real number.