Arithmetic Course/Differential Equation/Second Order Equation

Second Order Differential Equation

Second Ordered Differential Equation has the general form

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

Which can be expressed as

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

Solving 2nd Ordered Differential Equation

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

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

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

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

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

Case 1

$\lambda =0$

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

$s=-\alpha x$

$f(x)=e^{(}-\alpha x)$

Case 2

$\lambda >0$

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

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

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

$f(x)=Ae^{(}\alpha x)Cos\lambda x$

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

Case 3

$\lambda <0$

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

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

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

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

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