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

Answer:

Step-by-step 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.