Second Order Linear Homogeneous DE (Constant Coefficients)

Rarely Tested
Form: $$\frac{d^2y}{dx^2} + a\frac{dy}{dx} + by = 0$$
Auxiliary equation: $$m^2 + am + b = 0$$
Roots $$m_1, m_2$$:
  • Real & distinct: $$y = C_1 e^{m_1x} + C_2 e^{m_2x}$$
  • Real & equal: $$y = (C_1 + C_2 x)e^{m_1x}$$
  • Complex: $$m = \alpha \pm i\beta$$, solution: $$y = e^{\alpha x}(C_1 \cos \beta x + C_2 \sin \beta x)$$
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