Last modified on 6 May 2011, at 16:14

Arithmetic Course/Differential Equation/Second Order Equation

Second Order Differential EquationEdit

Second Ordered Differetial 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 EquationEdit

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 1Edit

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

Case 2Edit

\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 3Edit

\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)