Question: > 
> For the differential equation $\displaystyle a\frac{d^2x}{dt^2}+5\frac{dx}{dt}+5x=0,$ where $a>0$
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> Represent supercritical damping, Then range of $a$ as
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What i try :: Here we have campare with $\displaystyle A\frac{dx^2}{dt^2}+b\frac{dx}{dt}+C=0$

Then $A=a,B=5,C=5$

So here $D=B^2-4AC=25-5a$

Can anyone please tell me here $D>0$ or $D<0$ for supercritical damping.

Please help me , Thanks

Answer: 
There is a theorem that says that an integer $d$ is the "greatest common divisor" of two integers $a,b$ if and only if:

* $d$ is a divisor of both $a$ and $b$, i.e. $a=ud, b=vd$ for some integers $u, v$, and
* $d$ is a "linear combination" of $a$ and $b$, i.e. $d=xa+yb$ for some integers $x,y$.

So, knowing that $d=\gcd(a,b)$, to prove $md=gcd(ma, mb)$, you need to prove that $md$ is a common divisor of $ma, mb$ and that $md$ is a linear combination of $ma$ and $mb$.

* The first part is trivial: $a=ud$ implies $ma=u(md)$ and $b=vd$ implies $mb=v(md)$.
* The second part is what the proof you are referring to does: as $d=xa+yb$, obviously $md=x(ma)+y(mb)$, notably for *the same* $x, y$.
Hint: Instead of Bezout's, use the Euclidean algorithm on $(a,b)$ and on $(ma,mb)$ in parallel. (This necessarily has to be done somewhat abstractly, possibly with a smidge of induction.)