Question

The homogeneous part of the differential equation $\frac{d^{2}y}{d{x}^2}+p\frac{dy}{dx}+qy=r$ ($p,q,r$, are constants) has real distinct roots if

• A. $p^2-4q>0$
• B. $p^2-4q<0$
• C. $p^2-4q=0$
• D. $p^2-4q=r$

Answer 1

The correct option is A $p^2-4q>0$

Given $\frac{d^{2}y}{d{x}^2}+p\frac{dy}{dx}+qy=r$
$\Rightarrow (D^2+pD+q)y=r$
The auxiliary equation is
$m^2+pm+q=0$
$m=\frac{-p\pm \sqrt{p^2-4q}}{2}$
If $p^2-4q>0$, then the roots of AE are real and different.