Arithmetic Course/Differential Equation/Second Order Equation

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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)

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