Arithmetic Course/Differential Equation/Second Order Equation

Second Order Differential Equation

Second Ordered Differential Equation has the general form

$A\frac{d^{2}f(x)}{d{x}^2}+B\frac{df(x)}{dx}+C=0$

Which can be expressed as

$\frac{d^{2}f(x)}{d{x}^2}+\frac{B}{A}\frac{df(x)}{dx}+\frac{C}{A}=0$

Solving 2nd Ordered Differential Equation

$A\frac{d^{2}f(x)}{d{x}^2}+B\frac{df(x)}{dx}+C=0$

$\frac{d^{2}f(x)}{d{x}^2}+\frac{B}{A}\frac{df(x)}{dx}+\frac{C}{A}=0$

$s^2+\frac{B}{A}s+\frac{C}{A}=0$

$s=(-\alpha \pm \sqrt{{\alpha }^2-{\beta }^2})x$

$s=(-\alpha \pm \lambda )x$

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

$\lambda <0$

${\alpha }^2<{\lambda }^2$

$s=-\alpha x\pm j\lambda x$

$f(x)=e^{(}-\alpha x)[e^{(}\alpha x)+e^{(}-j\alpha x)]$

$f(x)=A{e}^{(}\alpha x)Sin\lambda x$

$A=\frac{1}{2j}{e}^{(}\alpha x)$