Auxillary Equation with real and dinstinct root

**Altair** · March 21st 2008, 03:33 AM

Please solve the following equation :-

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

We have only been taught to solve the equations of $y'$ or $y''$ sort. I managed to get the numerical value but I couldn't figure out that how did $t$ in $e^{(3/2)t}$ 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.

**mr fantastic** · March 21st 2008, 04:17 AM

Originally Posted by Altair

Please solve the following equation :-

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

We have only been taught to solve the equations of $y'$ or $y''$ sort. I managed to get the numerical value but I couldn't figure out that how did $t$ in $e^{(3/2)t}$ 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!

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

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

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

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