Arithmetic Course/Differential Equation/Second Order Equation

Second Order Differential Equation
Second Ordered Differential Equation has the general form
 * $$A \frac{d^2 f(x)}{dx^2} + B \frac{d f(x)}{dx} + C = 0 $$

Which can be expressed as
 * $$\frac{d^2 f(x)}{dx^2} + \frac{B}{A} \frac{d f(x)}{dx} + \frac{C}{A} = 0 $$

Solving 2nd Ordered Differential Equation

 * $$A \frac{d^2 f(x)}{dx^2} + B \frac{d f(x)}{dx} + C = 0 $$
 * $$\frac{d^2 f(x)}{dx^2} + \frac{B}{A} \frac{d f(x)}{dx} + \frac{C}{A} = 0 $$
 * $$s^2 + \frac{B}{A} s + \frac{C}{A} = 0 $$
 * $$s = (-\alpha \pm \sqrt{\alpha^2 - \beta^2}) x $$
 * $$s = (-\alpha \pm \lambda) x $$

Case 1

 * $$\lambda = 0$$
 * $$\alpha^2 = \lambda^2$$
 * $$s = -\alpha x$$
 * $$f(x) = e^(-\alpha x)$$

Case 2

 * $$\lambda > 0$$
 * $$\alpha^2 > \lambda^2$$
 * $$s = -\alpha x \pm \lambda x$$
 * $$f(x) = e^(\alpha x) [e^(-\alpha x) + e^(-\alpha x)]$$
 * $$f(x) = A e^(\alpha x) Cos \lambda x$$
 * $$A = \frac{1}{2} e^(\alpha x)$$

Case 3

 * $$\lambda < 0$$
 * $$\alpha^2 < \lambda^2$$
 * $$s = -\alpha x \pm j \lambda x$$
 * $$f(x) = e^(-\alpha x) [e^(\alpha x) + e^(-j\alpha x)]$$
 * $$f(x) = A e^(\alpha x) Sin \lambda x$$
 * $$A = \frac{1}{2j} e^(\alpha x)$$