Subtopic: Higher order linear ODEs with constant coefficients
Question Statement:
Consider the differential equation $\frac{d^2 y}{dx^2} + 5\frac{dy}{dx} + 6y = 0$. The product of the two characteristic roots of this equation is ________ (answer in integer).
Answer Key:
6
Step-by-Step Solution & Explanation
The auxiliary equation is:
$m^2 + 5m + 6 = 0$
Factoring the quadratic equation:
$(m + 2)(m + 3) = 0$
The roots are $m_1 = -2$ and $m_2 = -3$.
The product of the roots is:
$m_1 \cdot m_2 = (-2) \times (-3) = 6$
(By Vieta's formulas, for $a m^2 + b m + c = 0$, the product of roots is $c/a = 6/1 = 6$).