The differential equation
[tex]ay'' + by' + c = 0[/tex]
has characteristic equation
[tex]ar^2 + br + c = 0[/tex]
with roots [tex]r = \frac{-b\pm\sqrt{b^2-4ac}}{2a} = \frac{-b\pm\sqrt{D}}{2a}[/tex].
• If [tex]D>0[/tex], the roots are real and distinct, and the general solution is
[tex]y = C_1 e^{r_1x} + C_2 e^{r_2x}[/tex]
• If [tex]D=0[/tex], there is a repeated root and the general solution is
[tex]y = C_1 e^{rx} + C_2 x e^{rx}[/tex]
• If [tex]D<0[/tex], the roots are a complex conjugate pair [tex]r=\alpha\pm\beta i[/tex], and the general solution is
[tex]y = C_1 e^{(\alpha+\beta i)x} + C_2 e^{(\alpha-\beta i)x}[/tex]
which, by Euler's identity, can be expressed as
[tex]y = C_1 e^{\alpha x} \cos(\beta x) + C_2 e^{\alpha x} \sin(\beta x)[/tex]
The solution curve in plot (A) has a somewhat periodic nature to it, so [tex]\boxed{D < 0}[/tex]. The plot suggests that [tex]y[/tex] will oscillate between -∞ and ∞ as [tex]x\to\infty[/tex], which tells us [tex]\alpha>0[/tex] (otherwise, if [tex]\alpha=0[/tex] the curve would be a simple bounded sine wave, or if [tex]\alpha<0[/tex] the curve would still oscillate but converge to 0). Since [tex]\alpha[/tex] is the real part of the characteristic root, and we assume [tex]a>0[/tex], we have
[tex]\alpha = -\dfrac b{2a} > 0 \implies -b > 0 \implies \boxed{b < 0}[/tex]
Since [tex]D=b^2-4ac<0[/tex], we have
[tex]b^2 < 4ac \implies c > \dfrac{b^2}{4a} \implies \boxed{c>0}[/tex]
The solution curve in plot (B) is not periodic, so [tex]D\ge0[/tex]. For [tex]x[/tex] near 0, the exponential terms behave like constants (i.e. [tex]e^{rx}\to1[/tex]). This means that
• if [tex]D>0[/tex], for some small neighborhood around [tex]x=0[/tex], the curve is approximately constant,
[tex]y = C_1 e^{r_1x} + C_2 e^{r_2x} \approx C_1 + C_2[/tex]
• if [tex]\boxed{D=0}[/tex], for some small neighborhood around [tex]x=0[/tex], the curve is approximately linear,
[tex]y = C_1 e^{rx} + C_2 x e^{rx} \approx C_1 + C_2 x[/tex]
Since [tex]D=b^2-4ac=0[/tex], it follows that
[tex]b^2=4ac \implies c = \dfrac{b^2}{4a} \implies \boxed{c>0}[/tex]
As [tex]x\to\infty[/tex], we see [tex]y\to-\infty[/tex] which means the characteristic root is positive (otherwise we would have [tex]y\to0[/tex]), and in turn
[tex]r = -\dfrac b{2a} > 0 \implies -b > 0 \implies \boxed{b < 0}[/tex]
