CHARACTERISTIC EQUATIONS with
COMPLEX or REPEATED ROOTS

Consider equations of the form $L[y]=a\frac{d^2y}{dt^2}+ b\frac{dy}{dt}+ cy = 0,$
given y(t₀)=y₀, y'(t₀)=y'₀.

Complex Roots:

assume b²-4ac < 0.

• Solutions have the form y(t) = c₁e^r₁t + c₂e^r₂t,
  where $r_{1,2} = -\frac{b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a} = \lambda \pm i\mu$.
• Euler's Formula says $e^{it} = \cos(t) + i\sin(t)$, so
  $y(t) = e^{\lambda t} (\ c_1(\cos(\mu t)+i\sin(\mu t)) + c_2(\cos(\mu t)-i\sin(\mu t))\ )$.
• Real-valued solutions have the form
  $y(t) = e^{\lambda t}(\ c_1\cos(\mu t) + c_2\sin(\mu t)\ )$, $c_1,\ c_2$ real.
• Wronskian is $W(e^{\lambda t}\cos(\mu t),e^{\lambda t}\sin(\mu t)) = \mu e^{2\lambda t}$.

• Typical solutions are

  1.

  exponentially growing oscillations, when $\lambda > 0$, or

  2.

  exponentially decaying oscillations, when $\lambda < 0$, or

  3.

  steady oscillations, when $\lambda = 0$.

Repeated Roots:

assume b²-4ac = 0.

• One solution is y(t) = Ce^-bt/(2a); need two solutions.
• Try y(t) = v(t)e^-bt/(2a), then
  v''=0, so v(t) = c₁+c₂t.
• Solutions have the form y(t) = ( c₁ + c₂t)e^-bt/(2a).
• Wronskian is W(e^-bt/(2a),te^-bt/(2a)) = e^-bt/a.

Reduction of Order Method:

let y(t)=v(t)y₁(t).
If L[y₁]=0, then v' satisifies a first order equation

y₁(v')'+(2y'₁+py₁)v'=0.