## Elementary Differential Equations and Boundary Value Problems 9th Edition

a) For 2 real distinct negative roots, $b^2>4ac$ and $|\sqrt{b^2-4ac}|0$. b) For 2 real distinct roots with opposite signs, $b^2>4ac$ and $b-\sqrt{b^2-4ac}=-(b+\sqrt{b^2-4ac})\Rightarrow{b}=0,\quad{a}<0,\quad{c}>0\quad{or}\quad a<0,\quad{c}>0$ c) For 2 real distinct positive roots, $b^2>4ac$ and $|\sqrt{b^2-4ac}|$
$$ay''+by'+cy=d$$Let $y-e^{\lambda{x}}$ so that $(\ln{y})'=\lambda$. $a{\lambda}^2+b{\lambda}+c=0$ ($\because$ equilibrium solution causes LHS=0) $$\lambda_{1,2}=\frac{-b\pm{\sqrt{b^2-4ac}}}{2a}$$ As with root analysis in quadratic equations, we will aim to deduce conditions in a, b and c by studying the discriminant $D=b^2-4ac$. For 2 real distinct negative roots, $b^2>4ac$ and $|\sqrt{b^2-4ac}|0$. For 2 real distinct roots with opposite signs, $b^2>4ac$ and $b-\sqrt{b^2-4ac}=-(b+\sqrt{b^2-4ac})\Rightarrow{b}=0,\quad{a}<0,\quad{c}>0\quad{or}\quad a<0,\quad{c}>0$ For 2 real distinct positive roots, $b^2>4ac$ and $|\sqrt{b^2-4ac}|$