# Arithmetic Course/Differential Equation/Second Order Equation

## Second Order Differential Equation

Second Ordered Differential Equation has the general form

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

Which can be expressed as

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

## Solving 2nd Ordered Differential Equation

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

### Case 1

${\displaystyle \lambda =0}$
${\displaystyle \alpha ^{2}=\lambda ^{2}}$
${\displaystyle s=-\alpha x}$
${\displaystyle f(x)=e^{(}-\alpha x)}$

### Case 2

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

### Case 3

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