Calculus Course/Differential Equations/2nd Order Differential Equations

2nd Order Differential Equation

2nd Order Differential Equation is an equation that has the general form

$a \frac{d^2}{dx^2} f(x) + b \frac{d}{dx} f(x) + c = 0$

Characteristic Equation

2nd Order Differential Equations above can be rewritten as shown

$\frac{d^2 }{dx^2} f(x) + \frac{b}{a} \frac{d }{dx} f(x) + \frac{c}{a} = 0$

Let

$s = \frac{d}{dx}$

Then

$s^2 + \frac{b}{a} s + \frac{c}{a}= 0$

$s = (-\alpha \pm \sqrt{\lambda}$) t

$\alpha = \frac{b}{2a}$

$\beta = \frac{c}{a}$

$\lambda = \sqrt{\alpha^2 - \beta^2}$

Case 1

When

$\lambda = 0$

Then

$\alpha^2 = \beta^2$

$s = e^(-\alpha t)$

Equation has one real roots

Case 2

When

$\lambda > 0$

Then

$\alpha^2 > \beta^2$

$s = e^(-\alpha x) e ^ [\pm (\lambda x)]$

Equation has two real roots

Case 3

When

$\lambda < 0$

Then

$\alpha^2 < \beta^2$

$s = e^(-\alpha t) [e^(\pm j\lambda t)]$

Equation has two compex roots

Special Case

Case 1

Differential Equation of the form

$\frac{d^2 f(t)}{dt^2} + \lambda = 0$

$s^2 = -\lambda$

Roots of equation

$s = \pm j\sqrt{\lambda}$

Case 2

Differential Equation of the form

$\frac{d^2 f(t)}{dt^2} - \lambda = 0$

$s^2 = \lambda$

Roots of equation

$s = \pm \sqrt{\lambda}$

Summary

2nd Order Differential Equation

$\frac{d^2 }{dx^2} f(x) + \frac{b}{a} \frac{d }{dx} f(x) + \frac{c}{a} = 0$

has roots depend on the value of $\lambda$

1. $\lambda = 0 . f(x) = e^(-\alpha x)$
2. $\lambda > 0 . f(x) = e^(-\alpha x) e^(\pm \lambda x)$
3. $\lambda < 0 . f(x) = e^(-\alpha x) e^(\pm j\lambda x)$

With

$\alpha = \frac{b}{a}$

$\beta = \frac{c}{a}$

$\lambda = \sqrt{\alpha^2 - \beta^2}$