Differential equations -stable solutions?

Question

Which of the following IVP(initial value problem) is stable?
(i.e. the solutions converge for small changes
in the initial value.)

(1) y' - y = -4e^(-x) y(0)=2
(2) y' -2y = -6e^(-x) y(0)=2
(3) y' +2y = 2e^(-x) y(0)=2

Answer 1

you can determine graphically using runge-kutta methods:

http://en.wikipedia.org/wiki/Runge_kutta

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Answer 2

Solution
1) y=C e^x+ke^-x -k-1 =-4 so k=3 y= C*e^x+3e^-x
y(0) = 2+z y(0+z) = Ce^z +3e^-z which for z==>0 converges to C+3 = 2 if c= -1
2)y=Ce^2x+2e^-x
y(0+z) Ce^2z+2e^-z==> C+2 = 2 if C=0
y= Ce^-2x+2e^-x
y(0+z) =Ce^-2z+2e^-z which for ===>0 has limit C+2=2 for C=0
So all are stable