Why before $e^{x}$,the solution was not possible? we know the important role of exponential function in solving of ordinary differential equation.But the solution can be done by using  another function like $10^x$ or $2^x$.The example below shows this
If we have $$y''-4y'+3y=0$$
assume the $y=10^{mx}$
$y'=m10^{mx}log(10)$
$y''=m^210^{mx}log^2(10)$
substitute these in the O.D.E to get
$10^{mx}log^2(10)-4(10^{mx}log(10))+3(10^{mx})=0$
$m^210^{mx}log^2(10)-4(m10^{mx}log(10))+3(10^{mx})=0$
$m^2log^2(10)-4mlog(10)+3=0$
The characteristics equation can be solved to get the roots.
We see that the solution of differential equation can be by using $y=10^{mx}$
So Why before this not possible and why the solution must be by using $y=e^{mx}$
 A: You didn't give the full quote of what you read, but I suspect the author is not distinguishing between $e^x$ and $10^x$.  What you need to solve these problems is an exponential function that is defined on the reals.  Long ago you could imagine $10^x$ as a $1$ followed by $x$ zeros, but that only works if $x$ is a natural number.  If you extend $10^x$ to real $x$ it will let you solve the problem as you have shown.  I'm not sure it is easy to do the extension without going through $e$, natural logs, Taylor series, etc. so in that sense you can't solve the problem without $e^x$.  Certainly all those $\log (10)$'s in your solution mean you know something about $e$.
A: Some of the relevant history is discussed in The Number e.  It seems the curves $y = k a^x$ were discussed by Huygens in 1661, somewhat before the number $e$ itself was actually defined (and also before the concept of differential equations). 
A: Yes it is possible to solve the equations with $10^{mx}$ as you've shown.  However, you'll notice the many appearances of $\log 10$ in the answer.  This does not happen when $e^{mx}$ is used.  So, $e^{mx}$ is used, because it gives us the simplest answers.
