Calculus Course/Differential Equations/2nd Order Differential Equations

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2nd Order Differential Equation[edit]

2nd Order Differential Equation is an equation that has the general form

a \frac{d^2}{dx^2} f(x) + b \frac{d}{dx} f(x) + c = 0

Characteristic Equation[edit]

2nd Order Differential Equations above can be rewritten as shown

\frac{d^2 }{dx^2} f(x) + \frac{b}{a} \frac{d }{dx} f(x) + \frac{c}{a} = 0

Let

s = \frac{d}{dx}

Then

s^2 + \frac{b}{a} s + \frac{c}{a}= 0
s = (-\alpha \pm \sqrt{\lambda}) t
\alpha = \frac{b}{2a}
\beta = \frac{c}{a}
\lambda = \sqrt{\alpha^2 - \beta^2}

Case 1[edit]

When

\lambda = 0

Then

\alpha^2 = \beta^2
s = e^(-\alpha t)

Equation has one real roots

Case 2[edit]

When

\lambda > 0

Then

\alpha^2 > \beta^2
s = e^(-\alpha x) e ^ [\pm (\lambda x)]

Equation has two real roots

Case 3[edit]

When

\lambda < 0

Then

\alpha^2 < \beta^2
s = e^(-\alpha t) [e^(\pm j\lambda t)]

Equation has two compex roots

Special Case[edit]

Case 1[edit]

Differential Equation of the form

\frac{d^2 f(t)}{dt^2} + \lambda = 0
s^2 = -\lambda

Roots of equation

s = \pm j\sqrt{\lambda}

Case 2[edit]

Differential Equation of the form

\frac{d^2 f(t)}{dt^2} - \lambda = 0
s^2 = \lambda

Roots of equation

s = \pm \sqrt{\lambda}

Summary[edit]

2nd Order Differential Equation

\frac{d^2 }{dx^2} f(x) + \frac{b}{a} \frac{d }{dx} f(x) + \frac{c}{a} = 0

has roots depend on the value of \lambda

  1. \lambda = 0 . f(x) = e^(-\alpha x)
  2. \lambda > 0 . f(x) = e^(-\alpha x) e^(\pm \lambda x)
  3. \lambda < 0 . f(x) = e^(-\alpha x) e^(\pm j\lambda x)

With

\alpha = \frac{b}{a}
\beta = \frac{c}{a}
\lambda = \sqrt{\alpha^2 - \beta^2}