Solution to an ordinary differential equation The general solution to the equation $y''+by'+cy=0$ approaches to $0$ as $x$ approaches infinity if


*

*$b$ is negative $c$ is positive 

*$b$ is positive $c$ is negative 

*$b$ is positive $c$ is positive

*$b$ is negative $c$ is negative


To seek a non-zero solution, put $y=e^{mx}$ ($m$ is arbitrary constant) in the given ODE
Then the auxiliary equation you get is $e^{mx}(m^2+bm+c)=0$
or $m^2+bm+c=0$ as $e^{mx}\neq 0$. 
In case of real roots,what you want is that this quadratic  equation must not have a positive real root for which $b>0$ and $c>0$ (by Descarte's rule of signs) is sufficient.
In case no real root exists, even then you need the real part of the root to be negative for which you must have $b>0$. 
In both cases, option $3$ is the best choice.
