View source for Calculus Course/Differential Equations/2nd Order Differential Equations

==2nd Order Differential Equation==
2nd Order Differential Equation is an equation that has the general form
:<math>a \frac{d^2}{dx^2} f(x) + b \frac{d}{dx} f(x) + c = 0</math>

==Characteristic Equation==
2nd Order Differential Equations above can be rewritten as shown
:<math>\frac{d^2 }{dx^2} f(x) + \frac{b}{a} \frac{d }{dx} f(x) + \frac{c}{a} = 0</math>

Let 
: <math>s = \frac{d}{dx}</math>
Then
:<math>s^2 + \frac{b}{a} s + \frac{c}{a}= 0</math>

:<math>s = (-\alpha \pm \sqrt{\lambda}</math>) t
:<math>\alpha = \frac{b}{2a}</math>
:<math>\beta = \frac{c}{a}</math>
:<math>\lambda = \sqrt{\alpha^2 - \beta^2} </math>

===Case 1===
When 
:<math>\lambda = 0</math>
Then
:<math>\alpha^2 = \beta^2</math>
:<math>s = e^(-\alpha t)</math>
Equation has ''one real'' roots

===Case 2===
When
:<math>\lambda > 0</math>
Then
:<math>\alpha^2 > \beta^2</math>
:<math>s = e^(-\alpha x) e ^ [\pm (\lambda x)]</math>
Equation has ''two real'' roots

===Case 3===
When
:<math>\lambda < 0</math>
Then
:<math>\alpha^2 < \beta^2</math>
:<math>s = e^(-\alpha t) [e^(\pm j\lambda t)]</math>
Equation has ''two compex'' roots

==Special Case==
===Case 1===
Differential Equation of the form
:<math>\frac{d^2 f(t)}{dt^2} + \lambda = 0</math>
:<math>s^2 = -\lambda </math>

Roots of equation
:<math>s = \pm j\sqrt{\lambda} </math>
===Case 2===
Differential Equation of the form
:<math>\frac{d^2 f(t)}{dt^2} - \lambda = 0</math>
:<math>s^2 = \lambda </math>

Roots of equation
:<math>s = \pm \sqrt{\lambda} </math>

==Summary==
2nd Order Differential Equation
:<math>\frac{d^2 }{dx^2} f(x) + \frac{b}{a} \frac{d }{dx} f(x) + \frac{c}{a} = 0</math>
has roots depend on the value of <math>\lambda</math>
# <math>\lambda = 0 . f(x) = e^(-\alpha x) </math>
# <math>\lambda > 0 . f(x) = e^(-\alpha x) e^(\pm \lambda x)</math>
# <math>\lambda < 0 . f(x) = e^(-\alpha x) e^(\pm j\lambda x)</math>
With
: <math>\alpha = \frac{b}{a}</math>
: <math>\beta = \frac{c}{a}</math>
: <math>\lambda = \sqrt{\alpha^2 - \beta^2}</math>






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