# Thread: Auxillary Equation with real and dinstinct root

1. ## Auxillary Equation with real and dinstinct root

Please solve the following equation :-

$\displaystyle 2\ddot{x} + 5\dot{x} - 12x = 0$

We have only been taught to solve the equations of $\displaystyle y'$ or $\displaystyle y''$ sort. I managed to get the numerical value but I couldn't figure out that how did $\displaystyle t$ in $\displaystyle e^{(3/2)t}$ come ? What's the general equation of this type ? And please also solve the equation step by step as I want to know the real way of solving these type of equations.

2. Originally Posted by Altair
Please solve the following equation :-

$\displaystyle 2\ddot{x} + 5\dot{x} - 12x = 0$

We have only been taught to solve the equations of $\displaystyle y'$ or $\displaystyle y''$ sort. I managed to get the numerical value but I couldn't figure out that how did $\displaystyle t$ in $\displaystyle e^{(3/2)t}$ come ? What's the general equation of this type ? And please also solve the equation step by step as I want to know the real way of solving these type of equations.
This equation has exactly the same form as the ones you're familiar with!

$\displaystyle \ddot{x} \equiv \frac{d^2 x}{d t^2}$ and $\displaystyle \dot{x} \equiv \frac{dx}{dt}$.

Assume a solution of the form $\displaystyle x = A e^{mt}$.

$\displaystyle 2 m^2 + 5m -12 = 0 \Rightarrow (2m - 3)(m+4) = 0$ etc.