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

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 $$\lambda$$ With
 * 1) $$\lambda = 0 . f(x) = e^(-\alpha x) $$
 * 2) $$\lambda > 0 . f(x) = e^(-\alpha x) e^(\pm \lambda x)$$
 * 3) $$\lambda < 0 . f(x) = e^(-\alpha x) e^(\pm j\lambda x)$$
 * $$\alpha = \frac{b}{a}$$
 * $$\beta = \frac{c}{a}$$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2}$$